\section{Introduction}
Modern enumerative geometry is based on a series of compactifications of the moduli space of smooth embedded curves of genus $g$ and curve class $\beta\in H^+_2(X,\mathbb Z)$ in a smooth projective variety $X$, and on their virtual intersection theory. The boundary of M. Kontsevich's space of stable maps represents maps from nodal curves, including multiple covers and contracted components. The genus zero theory of projective space provides a smooth compactification with normal crossing boundary. In higher genus, instead, contracted subcurves and finite covers may give rise to boundary components of excess dimension. Even though this moduli space satisfies R. Vakil's Murphy's law, a desingularisation of the main component certainly exists after H. Hironaka's work on resolution of singularities. The main application of the methods developed in this paper is an \emph{explicit modular desingularisation} of the main component of the moduli space of stable maps to projective space in genus two and degree $d\geq3$.

 \begin{thm*}
 There exists a logarithmically smooth and proper DM stack $\pazocal{V\!Z}_{2,n}(\mathbb P^r,d)$ over $\mathbb C$, with locally free logarithmic structure (therefore smooth) and a birational forgetful morphism to $\M{2}{n}{\mathbb P^r}{d}^{\text{main}}$, parametrising the following data:
 \begin{itemize}[leftmargin=.5cm]
  \item a pointed admissible hyperelliptic cover \[\psi\colon (C,D_R,p_1,\ldots,p_n)\to(T,D_B,\psi(p_1),\ldots,\psi(p_n))\]
  where $C$ is a prestable curve of arithmetic genus two, $D_R$ and $D_B$ are length six (ramification and branch) divisors, and $(T,D_B,\psi(\mathbf{p}))$ is a stable rational tree;
  \item a map $f\colon C\to \mathbb P^r$;
  \item a bubbling destabilisation $C\leftarrow\widetilde C$ and a contraction $\widetilde C\to \overline C$ to a curve with Gorenstein singularities, such that $f$ factors through $\bar{f}\colon\overline C\to \mathbb P^r$, and the latter is non-special on any subcurve of $\overline C$.
 \end{itemize}
\end{thm*}

More generally, for any smooth projective variety $X$, we construct a proper moduli space $\pazocal{V\!Z}_{2,n}(X,\beta)$ admitting a perfect obstruction theory and defining \emph{reduced Gromov-Witten invariants}. We expect a simpler enumerative content compared to standard Gromov-Witten invariants, and that they satisfy the quantum Lefschetz principle in the case of projective complete intersections.

\subsection{Main and boundary components}
The moduli space of stable maps to $X$ (see \cite{Kontsevich}) represents $f\colon C\to X$, where $C$ is a \emph{prestable} (nodal and reduced) curve such that every rational component of $C$ contracted by $f$ contains at least three \emph{special points} (markings and nodes).

When $X=\mathbb P^r$, $f$ is equivalently determined by the data of a line bundle $L$ on $C$, and $r+1$ sections of $L$ that do not vanish simultaneously on $C$. Forgetting the sections we obtain a morphism:
\[\M{g}{n}{\mathbb P^r}{d}\to\mathfrak{P}ic_{\mathcal C_{g,n}/\mathfrak M_{g,n}}.\]
Obstructions lie in $H^1(C,L)$ \cite{WangJie}. This implies that $\M{0}{n}{\mathbb P^r}{d}$ is smooth. On the other hand, for $g\geq 1$, boundary components may arise where $L$ restricts to a special line bundle on a subcurve of $C$. When $d>2g-2$ we can identify a \emph{main component}: the closure of the locus of maps from a smooth source curve.

In genus one, the generic point of a boundary component has a contracted subcurve of genus one, together with $k\leq r$ tails of positive degree \cite{Vre}.

In genus two, two types of boundary phenomena occur:
\begin{itemize}[leftmargin=0.5cm]
 \item the minimal subcurve of genus two (\emph{core}) is contracted, or
 \item the restriction of $f$ to the core is the hyperelliptic cover of a line.
\end{itemize}
As an example of the second phenomenon, the main component of $\overline{\pazocal M}_2(\mathbb P^3,5)$ has dimension $20$; but the locus of maps from a reducible curve $C=Z\bigcup_q R$, with $Z$ of genus two covering a line two-to-one, and $R\simeq\mathbb P^1$ parametrising a twisted cubic, has dimension $21$.

We stress that $\overline{\pazocal M}_{2,n}(\mathbb P^r,d)^{\text{main}}$ has no natural modular interpretation, since it is obtained by taking the closure of the nice locus. Moreover, its singularities along the boundary can be nasty \cite{VakilMurphy}. In \cite{BCquartics} we draw the consequences of the present construction, analysing the locus of smoothable maps in $\overline{\pazocal M}_{2}(\mathbb P^2,4)$, a moduli space with more than twenty irreducible components.

\subsection{Good factorisation through a Gorenstein curve}
Our approach to the desingularisation of the main component takes off from a simple observation: a line bundle of degree at least $2g-1$ on a \emph{minimal Gorenstein} curve of genus $g$ has vanishing $h^1$; we have already used this implicitly for nodal curves in the discussion of the irreducible components of $\M{g}{n}{\mathbb P^r}{d}$. The Gorenstein assumption makes this fact a straightforward consequence of Serre duality. Minimality is a weaker notion than irreducibility, but it is needed to ensure that the line bundle has sufficiently positive degree on every subcurve.

This observation raises a natural question: is it possible to replace every $f\colon C\to\mathbb P^r$ with a ``more positive''/``less obstructed'' $\overline f\colon\overline C\to \mathbb P^r$ by ``contracting'' the higher genus subcurves on which $f^*\mathcal O_{\mathbb P^r}(1)$ is special? The answer is ``no'', since every such map is \emph{smoothable}, i.e. it can be deformed into a map from a smooth curve.\footnote{The answer could also be ``yes'': it has been shown by M. Viscardi in genus one that maps from Gorenstein curves satisfying certain stability conditions give rise to irreducible compactifications of $\pazocal{M}_{1,n}(\mathbb P^r,d)$ \cite{VISC}; yet, their deformation theory is hard to grasp, because such is the deformation theory of the singularities that are involved \cite{SMY2}.}

This, on the other hand, gives us a strategy to study the main component:
\begin{enumerate}[leftmargin=.5cm]
 \item\label{item:contraction} Construct a universal contraction $\mathcal C\to\overline{\mathcal C}$ to a Gorenstein curve, by collapsing the subcurves of $C$ on which $f$ has low degree. This Gorenstein curve will not be ready available on $\M{g}{n}{\mathbb P^r}{d}$, because the contraction map $\mathcal C\to\overline{\mathcal C}$ has moduli (called \emph{crimping spaces} in \cite{vdW}, and \emph{moduli of attaching data} in \cite{SMY2}); we first need to introduce a modification of $\M{g}{n}{\mathbb P^r}{d}$ along the boundary.
 \item\label{item:factorisation} Take only those maps that admit a factorisation through $\overline{\mathcal C}$: as we have discussed, these are all smoothable, so the moduli space is at the very least irreducible. It provides a birational model of $\M{g}{n}{\mathbb P^r}{d}^{\text{main}}$, with the advantage of admitting a modular interpretation.
\end{enumerate}
Moreover, this space is unobstructed over a base that can be assumed to be smooth in the low-genus examples that we have at our disposal: \cite{RSPW1}, where the base is a logarithmic modification of the moduli space of prestable curves; and this paper, where the base is a logarithmic modification of the moduli space of hyperelliptic admissible covers. Once such a moduli space is constructed, the proof of smoothness for target $\mathbb P^r$ is entirely conceptual; furthermore, the same methods can be employed to approach the study of different targets (e.g. products of projective spaces, toric varieties, flag varieties) and stability conditions (e.g. quasimaps).

While point \eqref{item:factorisation} was essentially established for us by D. Ranganathan, K. Santos-Parker, and J. Wise \cite{RSPW1}, point \eqref{item:contraction} is at all open for $g\geq 2$. In the present work, we make a hopefully meaningful step in this direction.

\subsection{Logarithmic geometry \& singularities}
The moduli space of prestable curves has a natural logarithmic structure induced by its boundary divisor, thus keeping track of the nodes and their smoothing parameters \cite{KatoF}. This induces a logarithmic structure on the moduli space of maps as well. It is a natural question whether the desingularisation of the main component can be achieved by means of a logarithmic modification; it is indeed the case in genus one \cite{RSPW1}.

\subsubsection{Augmenting the logarithmic structure: admissible covers} Our first most relevant finding is that, instead, in genus two it is necessary to enrich the logarithmic structure of the base by passing to a moduli space of admissible covers \cite{HarrisMumford}.

Every smooth curve of genus two is hyperelliptic, i.e. the canonical class induces a two-to-one cover of a line branched along six points. The hyperelliptic cover is essentially unique, but, when the curve becomes nodal and in the presence of markings, the uniqueness is lost. It can be restored by making it part of the moduli functor. The resulting space of admissible covers is as nice as the moduli space of curves, but it has the advantage of encoding the Brill-Noether theory of the curve in the logarithmic structure \cite{Mochizuki}. The necessity of such enrichment can be understood already by looking at the isolated Gorenstein singularities of genus two \cite{B}: there are two families of these, basically corresponding to the choice of either a Weierstrass point or a conjugate pair in the semistable model. In order to tell these two cases apart logarithmically, in the construction of $\mathcal C\to\overline{\mathcal C}$, we start by considering stable maps from the source of an admissible cover.

\subsubsection{Tropical canonical divisors} This points to our second finding. The line bundle giving the contraction $\mathcal C\to\mathcal{C}^\prime$ (the first step towards $\overline{\mathcal C}$) is a vertical twist of the relative dualising line bundle. The twist is indeed logarithmic, and the piecewise-linear function on $\operatorname{trop}(\mathcal C)$ determining it is nothing but a \emph{tropical canonical divisor} satisfying certain requirements of compatibility with the admissible cover, and in particular realisable \cite{MUW}. We think of these as truly high-genus phenomena.


\subsubsection{Non-isolated singularities} Finally, we underline that non-reduced curves appear as fibres of $\overline{\mathcal C}$, which requires a careful study of what we call \emph{tailed ribbons}, and further distinguishes our work from its genus one ancestor. This should not come as a surprise: first, non-reduced curves can still be Gorenstein if the nilpotent structure is supported along one-dimensional components, rather than isolated points; second, \emph{ribbons} were introduced in the '90s as limits of smooth canonical curves when the latter tend to the hyperelliptic locus in moduli \cite{Fong}. They appear naturally in our work at the intersection of the main component with the hyperelliptic ones. It is possibly less expected that they show up as well when the core is contracted, as a way of interpolating between isolated singularities whose special branches differ. The construction of $\overline{\mathcal C}$ is concluded by gluing in $\mathcal{C}^\prime$ the hyperelliptic cover of a genus two subcurve supported along a boundary divisor.

As a note for future investigation, we remark that we perform both the contraction $\mathcal C\to\mathcal C^{\prime}$ and the pushout $\mathcal C^{\prime}\to\overline{\mathcal C}$ in sufficiently general families. We wonder whether a pointwise construction might be possible, realising both steps as special instances of a more general pushout of logarithmic subcurves, in the spirit of S. Bozlee's \cite{Bozlee}.

\subsection{Relation to other work and further directions}

\subsubsection{Local equations and resolution by blowing up}\label{sec:desingbyblowingup}
It is always possible to find a local embedding of $\M{g}{n}{\mathbb P^r}{d}$ in a smooth ambient space by looking at:
\[\operatorname{Tot}(\pi_*L)^{\oplus r+1}\subseteq\operatorname{Tot}(\pi_*L\otimes\mathcal O_C(n))^{\oplus r+1}\]
over the Picard stack, where $\mathcal O_C(1)$ is a relative polarisation for the universal curve $\pi\colon\mathcal C\to\mathfrak{P}ic_{g,n}$ \cite[\S 3.2]{ToricQuasimaps}. If the polarisation is chosen carefully, the embedding is given by $\approx\! g$ local equations, repeated $r+1$ times. An approach to the desingularisation of the main component is by blowing up according to these local equations \cite{HL,HLderivedresolution,HLN}. It is close in spirit to the original construction of R. Vakil and A. Zinger \cite{VZ} (in particular, it involves an iterated procedure and a good deal of book-keeping), and it has the advantage of simultaneously ``desingularising'' the sheaves $\pi_*\mathcal L^{\otimes k}, k\geq 1,$ on the main component, making them into vector bundles. This in turn makes the theory of projective complete intersections accessible via torus localisation \cite{ZingerredCYhyp,PoparedCYci}.
 
Y.Hu, J.Li, and J.Niu \cite{HLN} carry out this strategy in genus two. Their desingularisation seems more efficient than ours, in the (vague) sense that the information encoded by the admissible cover \emph{far from the special branch} is useless. In \cite{HuNiuTwisted}, Hu and Niu reconstruct the blow-up by gluing projective bundle strata indexed by \emph{treelike structures} and \emph{rooted level trees}; these data are reminiscent of piecewise-linear functions on tropical curves, though missing both the slope and the metric data. These authors have already pointed out the similarities between their indexing set and the combinatorial data appearing in the moduli space of multi-scale differentials \cite{BCGGM}. This relation certainly deserves further attention: we think that canonical divisors on tropical curves could provide the right language to talk about it, and the geometry of Gorenstein curves could be the informing principle of further investigations. We expect a crucial ingredient for an all-genus desingularisation will be a good compactification of canonical curves.

\subsubsection{Computations in Gromov-Witten theory} Naive computations of what we may now expect to coincide with our reduced invariants have made seldom appearances in the literature: for example, with Zinger's enumeration of genus two curves with a fixed complex structure in $\mathbb P^2$ and $\mathbb P^3$ \cite{Zinger2}, and the computation of characteristic numbers of plane curves due to T. Graber, J. Kock, and R. Pandharipande \cite{KockGraberPandha}. To make the relation with the latter work precise, we should first extend our methods to the analysis of relative and logarithmic stable maps (compare with \cite{BNR} and \cite{RSPW2} in genus one).

$\pazocal{V\!Z}_{2,n}(X,\beta)$ is only the main component of a moduli space of aligned admissible maps $\mathcal{A}_{2,n}(X,\beta)$, which dominates $\overline{\pazocal{M}}_{2,n}(X,\beta)$ and is virtually birational to it. The virtual class of $\mathcal{A}_{2,n}(X,\beta)$ is expected to split as the sum of its main and boundary components; the contribution of the latter should be expressible in terms of genus zero and reduced genus one invariants via virtual pushforward \cite{ManolachePush}. This would deliver an extension of the Li-Zinger's formula to genus two (see \cite{LZ,ZingerStdVSRed,CoatesManolache}). Together with a localisation computation of reduced invariants, as mentioned in the previous section, this would provide an alternative proof of the genus two mirror theorem for the quintic threefold \cite{GJR,CGL}. Along the same lines, it would ease the computation of genus two Gromov-Witten invariants of Fano and Calabi-Yau complete intersections in projective space.

\subsubsection{Logarithmic maps to toric varieties and tropical realisability} In \cite{RSPW2}, Ranganathan, Santos-Parker, and Wise apply their techniques to desingularising the space of genus one logarithmic maps to a toric variety with respect to its boundary. As an application, they are able to solve the \emph{realisability problem} for tropical curves of genus one \cite{Speyer}: which such curves arise as the tropicalization of a smooth genus one curve in a torus? With Ranganathan, we are working towards a similar result in genus two - where, as far as we know, there is at the moment no reasonable guess as to what the full answer should be, although a good understanding of the moduli space of tropical curves has been reached \cite{CuetoMarkwig}.

\subsubsection{Birational geometry of moduli spaces of curves} In \cite{B}, the first author produces a sequence of alternative compactifications of $\pazocal M_{2,n}$ based on replacing genus one and two subcurves with few special points by isolated Gorenstein singularities. Although we do not discuss it here, the techniques developed in this paper also provide a resolution of the rational maps among these spaces. Moreover, the universal Gorenstein curve constructed in this paper unveils the possibility of defining new birational models of $\overline{\pazocal M}_{2,n}$ by including non-reduced curves as well - contrary to $\overline{\pazocal M}_{2,n}(m)$, these models should respect the $\mathfrak S_n$-symmetry in the markings. It would be interesting to compare them with \cite{PolishchukJohnson}, and to establish their position in the Hassett-Keel program \cite{SMY2,AFS1}.

\subsection{Plan of the paper} In \S \ref{sec:preliminaries} we establish some language and background material concerning \emph{logarithmic curves}, their tropicalization, and the use of piecewise linear functions on it; \emph{admissible covers}, and their logarithmic structure; and \emph{Gorenstein curves}, with both isolated singularities and non-reduced structures, including a number of useful properties and formulae, 
the classification of isolated singularities of genus one (due to Smyth) and two (due to the first author), and a novel study of the non-isolated singularities occurring in our work.

In \S \ref{sec:alignedadmissiblecovers} we introduce the key player: a subdivision of the tropical moduli space of weighted admissible covers based on aligning (ordering) the vertices of the tropical curve with respect to a piecewise linear function constructed from tropical canonical divisors. This subdivision induces a logarithmically \'etale model of the moduli space of weighted admissible covers. It is on this model that we are able to construct a universal family of Gorenstein curves. This process consists of two steps: first, a birational contraction $\mathcal C\to \mathcal{C}^\prime$; then, a pushout/normalisation $\mathcal{C}^\prime\to\overline{\mathcal C}$.

In \S \ref{sec:maps} we apply these methods to desingularise $\overline{\pazocal{M}}_{2,n}(\mathbb P^r,d)^\text{main}$.

\subsection{Notations and conventions} We work throughout over $\mathbb C$.

Curves will always be projective and $S1$, i.e. without embedded points, but they may be non-reduced. Subcurves are not supposed to be irreducible, but they are usually connected. We call \emph{core} the minimal subcurve of genus two. We may refer to subcurves $C^\prime\subseteq C$ that intersect the rest of $C$ and the markings in only one (resp. two) at worst nodal point (or marking) as \emph{tails} (resp. \emph{bridges}).

The \emph{dual graph} $\Gamma$ of a nodal curve $C$ has a vertex for every irreducible component, an edge for every node, and a leg for every marking (labelled or not); it is endowed with a genus function $g\colon V(\Gamma)\to\mathbb Z$, and the genus of the graph is given by:
\[g(\Gamma)=h^1(\Gamma)+\sum_{v\in V(\Gamma)}g(v).\]
The graph $\Gamma$ may be further weighted with a function $w\colon V(\Gamma)\to\mathbb Z$, that should be thought of as recording the degree of a line bundle on $C$.

A graph with no loops is called a \emph{tree}; its valence one vertices are called \emph{leaves} (sometimes, one of them plays a different role and it is therefore named the \emph{root}). A tree with only two leaves is a \emph{chain}.

A \emph{tropical curve} is a graph metrised in a monoid $\overline M$ (most basically, $\mathbb R_{\geq 0}$), i.e. a graph $\Gamma$ as above together with a length function $\ell\colon E(\Gamma)\to\overline M$ on the set of edges (legs are considered to be infinite, instead).

A \emph{logarithmic space} is denoted by $X=(\underline{X},M_X)$ with logarithmic structure $\alpha\colon M_X\to(\mathcal O_X,\cdot)$ (also simply indicated by exponential/logarithmic notation). $M_X$ is a sheaf in the \'{e}tale topology of $\underline{X}$, and $\overline M_X$ denotes the characteristic sheaf $M_X/\alpha^{-1}(\mathcal O^*_X)$.

A logarithmic space $X$ gives rise to a rational polyhedral cone stack $\operatorname{trop}(X)$ via \emph{tropicalization}. We abide to the rule that the tropicalization of a logarithmically smooth curve (say $C$), which is a tropical curve in the above sense, is denoted by the corresponding piecewise-linear character (in this case $\scalebox{0.8}[1.3]{$\sqsubset$}$).

We have quite a few families of curves; usually we adopt the following notation:
\begin{itemize}[leftmargin=.5cm]
 \item $\pi\colon C\to S$ will denote a prestable curve, $\widetilde{C}$ (with $\tilde\pi$) a partial destabilisation of $C$, $C^\prime$ (with $\pi'$) a (not necessarily Gorenstein) contraction of $\widetilde{C}$, and $\overline{C}$ (with $\bar\pi$) a (not necessarily reduced) Gorenstein curve dominated by $C^\prime$;
 \item $f\colon C\to X$ will denote a (stable) map to (a smooth and projective) $X$;
 \item $\psi\colon C\to T$ will denote an admissible cover from a genus two to a rational curve, $\psi'\colon C^\prime \to T^\prime$ a double cover (where the curves are not necessarily prestable).
\end{itemize}


\subsection{Acknowledgements} We are grateful to Dhruv Ranganathan and Jonathan Wise for nurturing this project. We would also like to thank Daniele Agostini, Fabio Bernasconi, Sebastian Bozlee, David Holmes, Yi Hu, Navid Nabijou, and Jingchen Niu for  helpful conversations, and Zak Tur\v{c}inovi\'{c} for his support with the pictures. We thank Cristina Manolache and Richard Thomas for pushing us in this direction during our doctoral studies. L.B. is supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy EXC-2181/1 - 390900948 (the Heidelberg STRUCTURES Cluster of Excellence). F.C. is supported by the starter grant “Categorified Donaldson–Thomas theory” No. 759967 of the European Research Council.
This work was initiated at the Max-Planck-Institut f\"ur Mathematik in Bonn; it was partly carried out during the workshop on ``Enumerative logarithmic geometry and mirror symmetry'' at the Mathematisches Forschungsinstitut Oberwolfach, and during a visit of the first author at the University of Edinburgh - we gratefully acknowledge all these institutions for their hospitality.

\section{Preliminaries and background material}\label{sec:preliminaries}
\subsection{Logarithmically smooth curves and their tropicalizations} Let $(S,M_S)$ be a logarithmic scheme. A family of \emph{logarithmically smooth curves} over $S$ is a proper and logarithmically smooth morphism $\pi\colon (C,M_C)\to (S,M_S)$ with connected one-dimensional (geometric) fibres, such that $\pi$ is integral and saturated. These hypotheses guarantee flatness and that the fibres are reduced. Logarithmically smooth curves naturally provide a compactification of the moduli space of smooth curves.

\subsubsection{Characterisation of logarithmically smooth curves}
F. Kato proved in \cite{KatoF} that logarithmically smooth curves have at worst nodal singularities; moreover, he provided the following local description: for every geometric point $p\in C$, there exists an \'etale local neighbourhood of $p$ in $C$ with a strict \'etale morphism to
\begin{description}
 \item[{smooth point}] $\mathbb A^1_S$ with the log structure pulled back form the base;
 \item[{marking}] $\mathbb A^1_S$ with the log structure generated by the zero section and $\pi^*M_S$;
 \item[{node}] $\mathcal O_S[x,y]/(xy=t)$ for some $t\in\mathcal O_S$, with semistable log structure induced by the multiplication map $\mathbb A^2_S\to\mathbb A^1_S$ and $t\colon S\to \mathbb A^1$.
\end{description}

\noindent In the last case, the class of $\log(t)$ in $\overline{M}_S$ is called a \emph{smoothing parameter} for the node.

\noindent At times, we may have to consider more general logarithmic orbicurves \cite{OlssonLogCurves}.

\subsubsection{Minimal logarithmic structures}
For every logarithmically smooth curve $C$ on a scheme $\underline{S}$, there is a \emph{minimal} logarithmic structure $M_S^\text{can}$ on $\underline{S}$ from which $C$ is pulled back; if $S=\{s\}$ is a geometric point, the characteristic sheaf of the minimal structure is free generated by the smoothing parameters of the nodes of $C_s$: \[\overline{M}^{\text{can}}_S=\mathbb N^{\#E(\Gamma(C_s))}.\]

By a result of W.D. Gillam \cite{Gillam}, the existence of minimal objects shows that the stack of logarithmically smooth curves over $(\text{LogSch})$ is the pullback of a stack over $(\text{Sch})$ endowed with a logarithmic structure: this is nothing but the Artin stack of prestable marked curves endowed with the logarithmic structure associated to its normal crossings boundary, which we are going to denote by $\mathfrak M_{g,n}^\text{log}$.

\subsubsection{Tropicalization of curves}
For a logarithmically smooth curve $(C,M_C)$ over a geometric point $(S,M_S)$, the \emph{tropicalization} $\scalebox{0.8}[1.3]{$\sqsubset$}$ of $C$ consists of its dual graph $\Gamma(C)$ metrised in $\overline{M}_S$; the length function $\ell\colon E(\Gamma(C))\to\overline{M}_S$ associates to each edge the smoothing parameter of the corresponding node. More precisely, vertices correspond to irreducible components weighted by their geometric genus, and there are (labelled) infinite legs corresponding to the markings. The tropicalization of $C$ is thus a family of classical (i.e. metrised in $\mathbb R$) tropical curves over the cone $\operatorname{Hom}(\overline{M}_S,\mathbb R_{\geq0})$. The construction can be generalised to more general base logarithmic schemes \cite[\S 7]{CCUW}, and it induces a (strict, smooth, surjective) morphism:
\[\mathfrak M_{g,n}^\text{log}\to\widetilde{\mathfrak M}_{g,n}^\text{trop},\]
where the latter is the lift of the stack of tropical curves to the category of logarithmic schemes through the tropicalization map.

\subsubsection{Piecewise-linear functions and line bundles}
The characteristic monoid at a node $q\in C$ is the amalgamated sum: \[\overline M_{C,q}=\overline M_{S,\pi(q)}\oplus_\mathbb N \mathbb N^{\oplus2},\]
where the map $\mathbb N\to\overline M_{S,\pi(q)}$ is the smoothing parameter $\delta_q$, and $\mathbb N\to\mathbb N^{\oplus 2}$ is the diagonal. It has been noticed in \cite{GS} that:
\[\overline M_{C,q}\simeq\{(\lambda_1,\lambda_2\in\overline M_{S,\pi(q)}^{\oplus2}|\lambda_2-\lambda_1\in\mathbb Z\delta_q\},\]
which, together with a similar analysis at the markings, motivates the observation \cite[Remark 7.3]{CCUW} that sections of $\overline M_C$ correspond to \emph{piecewise-linear} (PL) \emph{functions} on $\scalebox{0.8}[1.3]{$\sqsubset$}$ with values in $\overline M_S$. Similarly, every section of $\overline M_C/\pi^*\overline M_S$ is described by the collection of slopes of a rational function on $\scalebox{0.8}[1.3]{$\sqsubset$}$. The exact sequence:
\[0\to \mathcal O_C^\times\to M_C^\text{gp}\to\overline M_C^\text{gp}\to 0,\]
shows that to every section $\lambda\in\Gamma(C,\overline M_C^\text{gp})$ there is an associated $\mathcal O_C^\times$-torsor (or, equivalently, a line bundle) of lifts of $\lambda$ to $M_C^\text{gp}$. Over a geometric point (or, more generally, if $\overline M_S$ and $\scalebox{0.8}[1.3]{$\sqsubset$}$ are constant over $S$ - in particular, if no nodes are smoothed out), then every vertex $v$ of $\scalebox{0.8}[1.3]{$\sqsubset$}$ determines an irreducible component of $C$, and the restriction of $\mathcal O_C(-\lambda)$ to $C_v$ is given explicitly by \cite[Proposition 2.4.1]{RSPW1}:
\[\mathcal O_{C_v}(\lambda)\simeq \mathcal O_{C_v}(\sum s(\lambda,e_q)q)\otimes\pi^*\mathcal O_S(\lambda(v)),\]                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
where $s(\lambda,e_q)$ denotes the outgoing slope of $\lambda$ along the edge corresponding to $q$ (either a marking or a node of $C$).   

If we started from $\lambda\in\Gamma(C,\overline M_C)$,
the associated line bundle $\mathcal O_C(-\lambda)$ would come with a cosection $\mathcal O_C(-\lambda)\to \mathcal O_C$ (induced by the logarithmic structure). Such a cosection is not always injective, but when it is it defines an effective Cartier divisor on $C$; when it is not, it will vanish along components of $C$. Nonetheless, the association of this \emph{generalised effective Cartier divisor} to $\lambda$ behaves well under pullbacks, and in fact a useful point of view on logarithmic structures is to consider them as a functorial system of generalised effective Cartier divisors indexed by  $\overline M_C$ (see \cite{BV} for the details). See also \cite[p.9]{Bozlee} for a description in local coordinates.

\subsubsection{Alignments and blow-ups}\label{sec:logblowups}
The monoid $\overline M$ induces a partial order on $\overline M^{\text{gp}}$: \[m_1\leq m_2 \qquad \Leftrightarrow \qquad m_2-m_1\in\overline M.\]
Given a logarithmic scheme $(\underline S,M_S)$ with a logarithmic ideal $K=(m_1\ldots,m_h)$, the logarithmic subfunctor of $S$ defined by requiring that there is always a minimum among the $m_i,\ i=1,\ldots,h,$ - i.e. that the ideal is locally principal - is represented by the blow-up of $S$ in the ideal $\alpha(K)$ \cite[\S 3.4]{Santos-Parker}. Tropically, it corresponds to a subdivision of the cone $\operatorname{Hom}(\overline{M}_S,\mathbb R_{\geq0})$, and vice versa.

This simple observation has had many fruitful applications in moduli theory, starting from \cite{MarcusWise}.

\subsection{Admissible covers and their logarithmic structure}\label{sec:adm}

Admissible covers have been introduced by J. Harris and D. Mumford in \cite{HarrisMumford} as a compactification of Hurwitz' spaces. A fully-fledged moduli theory for them has been developed only later by S. Mochizuki with the introduction of logarithmic techniques \cite{Mochizuki}, and by D. Abramovich, A. Corti, and A. Vistoli with the introduction of twisted curves \cite{ACV}. They have also been generalised by B. Kim in \cite[\S\S 5.2 and 7.2]{KimLog} (he calls them log stable $\mu$-ramified maps). We will only be concerned with the case of double covers of $\mathbb P^1$, i.e. hyperelliptic curves.

In order to motivate the construction, we observe that every smooth curve of genus two is hyperelliptic (under the canonical map) with six Weierstrass points (i.e. ramification points for the hyperelliptic cover; this follows from the Riemann-Hurwitz formula). When the curve is allowed to become nodal, there still exists a degree two map to the projective line, but it is no longer finite. In this case, it is appropriate to define Weierstrass points to be limit of Weierstrass points in a smoothing family; the drawback is that whether a point on a (contracted) rational subcurve is Weierstrass or not may depend on the choice of the smoothing family. To resolve this ambiguity within the definition of the moduli problem, a solution is to expand the target, so as to consider the space of \emph{finite} morphisms from nodal curves of genus two to rational \emph{trees}. Here is a formal definition.

\begin{dfn}\label{def:admissiblecover}
 A family of \emph{admissible hyperelliptic covers} over $S$ is a finite morphism $\psi\colon (C,D_R)\to (T,D_B)$ over $S$ such that:
 \begin{enumerate}
  \item $(C,D_R)$ and $(T,D_B)$ are prestable curves with (unlabelled) smooth disjoint multisections $D_R$ and $D_B$ of length $2g+2$, $C$ has arithmetic genus $g$, and $(T,D_B)$ is a stable rational tree;
  \item $\psi$ is a double cover on an open $U\subseteq T$ dense over $S$;
  \item\label{point:logetale} $\psi$ is \'etale on $C^\text{sm}\setminus D_R$, it maps $D_R$ to $D_B$ with simple ramification, and it maps nodes of $C$ to nodes of $T$ so that in local\footnote{We refer the reader to \cite{Mochizuki}, and in particular Remark 2 of \S3.9, for a detailed discussion.} coordinates:
  \[ \psi^\#\colon \mathcal O_S[u,v]/(uv-s)\to\mathcal O_S[x,y]/(xy-t)\]
  maps $u\mapsto x^i,v\mapsto y^i, s\mapsto t^i$ for $i=1$ or $2$.
 \end{enumerate}
\end{dfn}
Mochizuki shows that condition \eqref{point:logetale} can be replaced by requiring that $\psi$ lifts to a \emph{logarithmically \'etale} morphisms of logarithmic schemes $(C,M_C)\to(T,M_T)$ over $(S,M_S)$, so that the image of a smoothing parameter of $T$ is either a smoothing parameter of $C$ or its double. Moreover there is a minimal logarithmic structure over $S$ that makes this possible: if $p$ is a node of $C$ that $\psi$ maps to the node $q$ of $T$, \[M_S^{\psi-\text{can}}=M_S^{C-\text{can}}\oplus_{\mathcal O_S^*}M_S^{T-\text{can}}/\sim,\qquad (0,\delta_q)\sim(i\delta_p,0),\]
where $i$ is the local multiplicity defined at the end of \eqref{point:logetale}. In the case of double covers, this simply means that the minimal logarithmic structure for the admissible cover is the same as that of the source curve, except that the smoothing parameters of two nodes have been identified if $\psi$ matches them both with the same node of the target. We thus obtain a logarithmic Deligne-Mumford stack.
\begin{thm}[{\cite[\S 3.22]{Mochizuki}}] The moduli stack of admissible hyperelliptic covers $\mathcal A_{g,0,2}$ is a logarithmically smooth with locally free logarithmic structure (and therefore smooth), proper Deligne-Mumford stack, with a logarithmically \'etale morphism to $\overline{\pazocal M}_{0,2g+2}/\mathfrak S_{2g+2}$.
\end{thm}
To an admissible hyperelliptic cover we can associate a harmonic morphism of degree two to a metric tree $\psi\colon\scalebox{0.8}[1.3]{$\sqsubset$}\to\scalebox{0.9}[1.2]{$\top$}$. The tropical geometry of hyperelliptic and admissible covers has been analysed, for instance, in \cite{ChanHyperelliptic,CMR}.

\subsection{Gorenstein curves}
\subsubsection{Tools and formulae}
Curves shall always be assumed Cohen-Macaulay, i.e. $S1$; they might still be non-reduced along some subcurve, but they have no embedded points.


\begin{dfn}
 A curve $X$ is Gorenstein if its dualising sheaf $\omega_X$ is a line bundle.
\end{dfn}

A fundamental role in the study of singularities is played by the conductor ideal.
\begin{dfn}
 Let $\nu\colon\widetilde X\to X$ be a finite morphism. The \emph{conductor ideal} of $\nu$ is: \[\mathfrak c = \operatorname{Ann}_{\mathcal O_X}\left(\nu_*\mathcal O_{\widetilde X}/\mathcal O_X\right)=\mathcal{H}om_{X}(\nu_*\mathcal O_{\widetilde X},\mathcal O_X).\]
\end{dfn}
The conductor is the largest $\mathcal O_X-$ideal sheaf that is also a $\nu_*\mathcal O_{\widetilde X}-$ideal sheaf. When $X$ is a reduced curve with finite normalisation - it is always the case over a field -, blowing up the conductor ideal recovers the normalisation \cite{Wilson}. The conductor ideal admits a further characterisation in terms of duality theory as:
\[\mathfrak c =\nu^!\mathcal O_X=\omega_\nu.\]

\begin{rem}
For non-reduced curves, we abuse notation by calling normalisation of $X$ a finite birational morphism with source a possibly disconnected, non-reduced curve, whose underlying reduced curve is smooth . In other words, the source of the normalisation is a union of \emph{multiple curves} in the sense of \cite{Drezet-parametrisation}.
\end{rem}

The conductor ideal relates the dualising line bundle of a nodal curve and that of its normalisation; more generally \cite[Proposition 1.2]{Catanese}:

\begin{prop}[Noether's formula]\label{prop:Noether}
 Let $\nu\colon\widetilde X\to X$ be a finite birational morphism of Gorenstein curves. Then, viewing  $\mathfrak c$ as an ideal sheaf on $\widetilde X$: \[\omega_{\widetilde X}=\nu^*\omega_X\otimes\mathfrak c.\]
\end{prop}

\begin{cor}\label{cor:principalconductor}
 Let $X$ be a Gorenstein curve with Gorenstein normalisation $\nu\colon\widetilde X\to X$. The conductor is a principal ideal on $\widetilde X$.
\end{cor}

\begin{dfn}
 For a finite birational morphism $\nu\colon\widetilde{X}\to X$ of curves, the coherent $\mathcal O_X-$module $\nu_*\mathcal O_{\widetilde X}/\mathcal O_X$ has finite support; its length is called the \emph{$\delta$-invariant} of $\nu$. When $\nu$ is the normalisation, we simply call it the $\delta$-invariant of $X$. When $X$ is reduced, it is the sum of the local contributions of all the isolated singularities of $X$.
\end{dfn}

We review a result of J.P. Serre \cite[\S~4, Proposition~7]{serreAG} for possibly non-reduced curves.

\begin{lem}\label{lem:n=2delta}
 If $X$ is a Gorenstein curve and $\widetilde X$ its normalisation, then 
 \begin{equation}\label{eqn:Gorenstein}
  \operatorname{length}\left(\mathcal O_{\widetilde X}/\mathfrak c\right)=2\delta.
 \end{equation}
\end{lem}
\begin{proof}
Since the normalisation is finite and birational, we may work locally around the support of $\nu_*\mathcal O_{\widetilde X}/\mathcal O_X$, which we may therefore assume is a single point. Let $X=\operatorname{Spec}(R)$ and $\widetilde X=\operatorname{Spec}(\widetilde R)$. Using $\mathfrak c\subseteq R\subseteq \widetilde R$ we see that \eqref{eqn:Gorenstein} is equivalent to:
\[\dim_{\mathbb C}(R/\mathfrak c)=\delta.\]

Let $\omega$ be a local generator of $\Omega_R$; given $f\in R$, Noether's formula implies that $f\omega\in \Omega_{\widetilde{R}}$ if and only if $f\in \mathfrak c$.
We therefore get: \[R/\mathfrak c\hookrightarrow \Omega_R/\Omega_{\widetilde{R}}.\] Since $\Omega_R$ is free on rank one and $\omega$ a generator, the above map is surjective as well. Finally, applying $\operatorname{Hom}_R(-,\Omega_R)$ to the normalisation exact sequence, we obtain $\Omega_R/\Omega_{\widetilde{R}}=\operatorname{Ext}^1_R(\widetilde{R}/R,\Omega_R)$, and therefore $\dim_{\mathbb C}\Omega_R/\Omega_{\widetilde{R}}=\delta$.
\end{proof}


\begin{rem} The converse is true when $X$ is reduced, but may fail otherwise; an example of which is provided by the subalgebra of $\mathbb C[\![s,\epsilon]\!]\times \mathbb C[\![t]\!]$ generated by $\langle s,\epsilon,t\rangle$, i.e. the transverse union of a ribbon with a line, which, though satisfying \eqref{eqn:Gorenstein}, is not Gorenstein as a consequence of the following lemma. What seems to be lacking in the non-reduced case is a good theory of dualising sheaves as rational differential forms on the normalisation satisfying some residue condition.
\end{rem}

\begin{dfn}
 A curve is \emph{decomposable} if it is obtained by gluing two curves along a (reduced closed) point.
\end{dfn}
\begin{rem}
 This is equivalent to the definition of \cite[Definition 2.1]{Stevens}. Indeed, the fiber product over $\mathbb C$ of $\mathbb C[x_1,\ldots,x_m]/I_1$ and $\mathbb C[y_1,\ldots,y_n]/I_2$ is isomorphic to
 \begin{equation}\label{eqn:deomposable}
  \underbrace{\mathbb C[x_1,\ldots,x_m,y_1,\ldots,y_n]}_S/\underbrace{(I_1(\mathbf x),y_1,\ldots,y_n)}_{J_1}\cap\underbrace{(x_1,\ldots,x_m,I_2(\mathbf y))}_{J_2}.
 \end{equation}
On the other hand, with notation as in \eqref{eqn:deomposable}, there is a short exact sequence:
\[0\to S/J_1\cap J_2\to S/J_1\times S/J_2\to S/J_1+J_2\simeq\mathbb C\to 0,\]
which is exact in the middle because every element there can be represented as a pair of polynomials in $\mathbf x$ only (resp. in $\mathbf y$ only), which is easy to lift.
\end{rem}

We review \cite[Proposition 2.1]{AFSGm} for non-necessarily reduced curves.

\begin{lem}\label{lem:indecomposable}
 A decomposable curve may be Gorenstein only if it is a node.
\end{lem}
\begin{proof}
Let us assume that $X$ is the decomposable union of $X_1$ and $X_2$.

Notice first that if $X$ is Gorenstein, then so are $X_1$ and $X_2$.
Indeed, say we can find $(a_1,\ldots,a_n)\subseteq\mathfrak m_X$ a regular sequence such that $\mathcal O_{X^0}=\mathcal O_X/(a_1,\dots,a_n)$ is Gorenstein of dimension zero, i.e.  $\operatorname{Hom}_{X^0}(\mathbb C,\mathcal O_{X^0})\cong \mathbb C$ (and all higher Ext groups vanish, which is automatic by \cite[Theorem~18.1]{matsumura1989}).
Since $\mathfrak m_X= \mathfrak m_{X_1}\oplus\mathfrak m_{X_2}$, this defines a regular sequence in the latter as well, and  $\mathcal O_{X_1}\oplus \mathcal O_{X_2}/(a_1,\dots,a_n)\cong \mathcal O_{X_1^0}\oplus \mathcal O_{X_2^0}$ is an extension of $k$ with $\mathcal O_{X^0}$. From this we see that $\dim_{\mathbb C} \operatorname{Hom}_{X_i^0}(\mathbb C, \mathcal O_{X_i^0})=1,\ i=1,2,$ which is to say that $X_1$ and $X_2$ are Gorenstein as well.

Now, $X$ and $X_1\sqcup X_2$ have the same normalisation $\widetilde X$, and
\begin{equation}\label{eqn:deltacomparison}
\delta_{X}=\delta_{X_1}+\delta_{X_2} +1.  
\end{equation}
On the other hand, $\mathcal O_{\widetilde X}/\mathcal O_X\twoheadrightarrow \mathcal O_{\widetilde X}/\mathcal O_{X_1\sqcup X_2}$ implies that $\mathfrak c_X\subseteq\mathfrak c_{X_1\sqcup X_2}$ (as ideals of $\mathcal O_{\widetilde X}$).
We claim that, if $X$ is Gorenstein, the reduced curves underlying $X_1$ and $X_2$ are both smooth.
\begin{itemize}[leftmargin=.5cm]
 \item If they are both singular, then $\mathfrak c_{X_1\sqcup X_2}\subseteq \mathfrak m_{X_1}\oplus\mathfrak m_{X_2}=\mathfrak m_X$, so $\mathfrak c_X=\mathfrak c_{X_1\sqcup X_2}$. By Lemma \ref{lem:n=2delta}, we obtain:
 $2\delta_X=\dim_{\mathbb C}(\mathcal O_{\widetilde X}/\mathfrak c_X)=\dim_{\mathbb C}(\mathcal O_{\widetilde X}/\mathfrak c_{X_1\sqcup X_2})=2\delta_{X_1}+2\delta_{X_2}$, which contradicts \eqref{eqn:deltacomparison}.
 \item If $X_1$ is singular and $X_2$ smooth, \eqref{eqn:deltacomparison} reduces to $\delta_X=\delta_{X_1}+1$. On the other hand, $\mathfrak c_{X_2}=\mathcal O_{X_2}$, and the contradiction comes from:
 $2\delta_X=\dim_{\mathbb C}(\mathcal O_{\widetilde X}/\mathfrak c_X)=\dim_{\mathbb C}(\mathcal O_{\widetilde X}/\mathfrak c_{X_1})=2\delta_{X_1}$.
\end{itemize}

Finally, since both underlying curves are smooth, we see from the definition of decomposability that:
\[\widehat{\mathcal O}_X=\mathbb C[\![x,\epsilon_i,y,\eta_j]\!]/(xy,x\eta_j,\epsilon_i y,\epsilon_i\eta_j,\epsilon_i^{m_i},\eta_j^{n_j})_{\substack{i=1,\ldots,h\\j=1,\ldots,k}}.\]
If $X_1$ and $X_2$ are reduced, we recover the node. In all other cases, one can verify that $\dim_{\mathbb C}\operatorname{Hom}(\mathbb{C},\mathcal O_X)\geq 2$ (it is generated by $xy,x\epsilon_1^{m_1-1},\ldots$).

\end{proof}


\subsubsection{Isolated singularities}

Let $(X,x)$ be (the germ of) a reduced curve with a unique singular point $x$, with normalisation $\nu\colon\widetilde{X}\to X$. The following is a measure of how much of the arithmetic genus of a projective curve is hiding in its singularities.
\begin{dfn}\label{def:genus}\cite{SMY1}
If $X$ has $m$ branches at $x$, the \emph{genus} of $(X,x)$ is:
\[g=\delta-m+1.\] 
\end{dfn}

The classification of isolated Gorenstein singularities of genus one has been carried out by D.I. Smyth in \cite[A.3]{SMY1}.

\begin{prop}
 An $(X,x)$ of genus one with $m$ branches is locally isomorphic to:
\begin{description}
 \item[$m=1$] the cusp, $V(y^2-x^3)\subseteq\mathbb A^2_{x,y}$;
 \item[$m=2$] the tacnode, $V(y^2-yx^2)\subseteq\mathbb A^2_{x,y}$;
 \item[$m\geq 3$] the union of $m$ general lines through the origin of $\mathbb A^{m-1}$.
\end{description}
\end{prop}

All of these singularities are smoothable. Choosing a one-parameter smoothing and passing to a regular semistable model, the \emph{semistable tail}, i.e. the subcurve contracted to the singularity, admits a simple description: if we mark the semistable tail by the intersection with the rest of the central fibre, then it is a \emph{balanced} nodal curve of genus one. This means that it consists of a genus one core - which can be either smooth, or a circle of rational curves - together with some rational trees supporting the markings, and the distance between a marking and the core - i.e. the length of the corresponding rational chain - is independent of the chosen marking \cite[Proposition 2.12]{SMY1}.

\smallskip

The classification of isolated Gorenstein singularities of genus two has been carried out by the first author in \cite[\S2]{B}.

\begin{prop}
 The unique unibranch Gorenstein singularity of genus two is the \emph{ramphoidal cusp} or $A_4$-singularity, $V(y^2-x^5)\subseteq\mathbb A^2_{x,y}$. For every $m\geq 2$, there are exactly two  isomorphism classes of germs of isolated Gorenstein singularities of genus two:
\end{prop}

 
 \begin{tabular}{c||c|c}
  & type I & type II \\
  \hline
  \hline
  Parametr. & 
  \begin{minipage}{.37\textwidth}{
 \begin{align*}
 \begin{split}
  x_1= & t_1\oplus0\oplus\ldots\oplus t_m^3\\
  x_2= & 0\oplus t_2\oplus\ldots\oplus t_m^3\\
  &\ldots\\
  x_{m-1}= & 0\oplus\ldots\oplus t_{m-1}\oplus t_m^3\\
  x_m= & 0\oplus\ldots\oplus0\oplus t_m^2
  \end{split}
 \end{align*}}
  \end{minipage}
&
  \begin{minipage}{.39\textwidth}{
   \begin{align*}
 \begin{split}
  x_1= & t_1\oplus0\oplus\ldots\oplus t_m\\
  x_2= & 0\oplus t_2\oplus\ldots\oplus t_m^2\\
  &\ldots\\
  x_{m-1}= & 0\oplus\ldots\oplus t_{m-1}\oplus t_m^2\\%\pmod{\widetilde{\mathfrak m}^3}.
  (y= & 0\oplus t_2^3 \quad\text{if } m=2)
 \end{split}
 \end{align*}}
  \end{minipage}
\\
\hline
  Equations & 
  \begin{minipage}{.37\textwidth}{
  
  \begin{description}
  \item[{\textcolor{gray}{$m=1$}}] \textcolor{gray}{$x^5-y^2$ ($A_4$);}
  \item[$m=2$] $x_2(x_2^3-x_1^2)$ ($D_5$);
  \item[$m=3$] $\langle x_3(x_1-x_2),x_3^3-x_1x_2\rangle$;
  \item[$m\geq 4$] $\langle x_i(x_j-x_k),\\ x_m(x_i-x_j),\\ x_m^3-x_1x_2\rangle_{i\neq j\neq k\in\{1,\ldots,m-1\}}$
   \end{description}} 
  \end{minipage}
&
  \begin{minipage}{.39\textwidth}{
  
  \begin{description}
  \item[$m=2$] $y(y-x_1^3)$ ($A_5$);
  \item[$m=3$] $x_1x_2(x_2-x_1^2)$ ($D_6$);
  \item[$m\geq 4$] $\langle x_3(x_1^2-x_2),\\ x_i(x_j-x_k)\rangle_{\substack{1\leq i<j<k\leq m-1 \text{ or }\\1<j<k<i\leq m-1}}$
   \end{description}} 
  \end{minipage}
 \end{tabular}

\smallskip 
 
It is not hard to see that, for $m\geq 3$, every type I singularity is the union of a cusp with $m-1$ lines, and every type II singularity is the union of a tacnode with $m-2$ lines; in each case, we refer to the components of the genus one subcurve as the \emph{special branches}.

The description of semistable tails is a bit more cumbersome, but enlightening: assuming that the genus two core is smooth, the special branches are closer to the core and attached to a Weierstrass point (respectively, two conjugate points) in type I (resp. II); all other branches are equidistant and further away from the core. In fact, the ratio between the length of the special rational chain and the others is fixed to $\frac{1}{3}$ (resp. $\frac{1}{2}$) in type I (resp. II). For a more detailed statement in case the core is not smooth see \cite[Propositions 4.3-4.6]{B}.


\subsubsection{Non-reduced structures}

Multiple curves were investigated in the '90s in connection to Green's conjecture \cite{Bayer-Eisenbud}. Ribbons, in particular, were understood to arise as limits (in the Hilbert scheme of $\mathbb P^{g-1}$) of canonical curves, as the curve becomes hyperelliptic \cite{Fong}.

\begin{dfn}
 A \emph{ribbon} is a double structure on $\mathbb P^1$, i.e. a non-reduced curve $R$ with $R_{\text{red}}=\mathbb P^1$ defined by a square-zero ideal $\pazocal I_{R_{\text{red}}/R}$ that is a line bundle on $R_{\text{red}}$.
\end{dfn}


\begin{exa}
 There is only one ribbon of genus two, $R_2$, up to isomorphism: it is the first infinitesimal neighbourhood of the zero section in $\operatorname{Tot}_{\mathbb P^1}(\mathcal O(3))$. The short exact sequence
 \[0\to \mathcal I_{\mathbb P^1/R_2}\simeq\mathcal O_{\mathbb P^1}(-3)\to \mathcal O_{R_2}\to\mathcal O_{\mathbb P^1}\to 0\]
 is split by restricting the projection $\operatorname{Tot}_{\mathbb P^1}(\mathcal O(3))\to\mathbb P^1$ to $R_2$. $R_2$ is therefore called a \emph{split ribbon}. In fact, all ribbons of genus at most two are split.
\end{exa}

Automorphisms and moduli of multiple curves have been studied in \cite{Drezet-parametrisation}. We are going to encounter non-reduced structures along singular curves as well.

\begin{exa}\label{exa:1-tailedribbon}
 The simplest example of a Gorenstein non-reduced structure with singular underlying curve is given by the union of a ribbon and a line \emph{along a double point}. Local equations are $\mathbb C[\![x,y]\!]/(x^2y)$. It is easy to see that, for such a curve to have genus two, the ribbon needs to have ideal $\mathcal I_{\mathbb P^1/R}\simeq\mathcal O_{\mathbb P^1}(-2)$. Such a curve can be realised in the linear system $\lvert 2D_+ + F\rvert$ on the Hirzebruch surface $\mathbb F_2$, where $D_+$ denotes the class of the positive section, and $F$ the class of a fiber.
\end{exa}

\begin{exa}\label{exa:ribbonchain}
 Two ribbons $R_1$ and $R_2$ can be joined together with local equations $\mathbb C[\![x,y]\!]/(x^2y^2)$. If the whole curve has genus two, the sum of $\chi(\mathcal O_{R_1})$ and $\chi(\mathcal O_{R_2})$ is $3$. Thus, there is both an asymmetry and some ambiguity as to which partition of $3$ we shall see. We justify below that $3=2+1$ is the correct one.
\end{exa}

\begin{exa}\label{exa:tailedribbons}
 More generally, the union of a ribbon with a rational $k$-fold point along a double point (representing a generic tangent vector to the $k$-fold point) is Gorenstein. Local equations are given by
 \[\mathbb C[\![x_1,\ldots,x_k,y]\!]/(x_ix_j,(x_i-x_j)y)_{1\leq i <j\leq k}.\]
 To see that this is Gorenstein, it is enough to find a regular element $\xi$ such that the quotient be Gorenstein of dimension zero. Let $\xi=\sum_{i=1}^kx_i-y$. The quotient $R$ is a graded finite-dimensional algebra, with $R_0=\mathbb C$, $R_1=\mathbb C\langle x_1,\ldots,x_k\rangle$, and $R_2=\mathbb C\langle x_1^2=\ldots=x_k^2\rangle$, having one-dimensional socle $R_2$.
 
 Now, we can obtain a Gorenstein projective curve of genus two, $C$, by gluing a ribbon together with some $k_i$-fold points ($i=1,\ldots,r$) at distinct (closed) points of the ribbon, by iterating the local construction above. From the short exact sequence
 \[0\to \mathcal O_C\to \mathcal O_R\oplus\bigoplus_{i=1}^r\mathcal O_{\mathbb P^1}^{\oplus k_i}\to \bigoplus_{i=1}^r(\mathbb C^{k_i-1}\oplus\mathbb C[\epsilon])\to 0,\]
 it is easy to see that the ribbon $R$ should fit in
 \[0\to\mathcal O_{\mathbb P^1}(r-3)\to\mathcal O_R\to\mathcal O_{\mathbb P^1}\to0,\]
 depending only on the total number of ``noded'' points $r$, and not on the number of branches of each $k_i$-fold point.
\end{exa}
\begin{dfn}
 We call $C$ as in Example \ref{exa:tailedribbons} a \emph{$(k_1,\ldots,k_r)$-tailed ribbon}.
\end{dfn}

\begin{rem}\label{rem:Noetheradjunction}
 We can employ Noether's formula (Proposition \ref{prop:Noether}) and adjunction (on a surface containing the ribbon, i.e. $\operatorname{Tot}(\mathcal O_{\mathbb P^1}(3-r))$) to compute the restriction of the dualising sheaf to every component of a $(k_1,\ldots,k_r)$-tailed ribbon $C$. First, the ``normalisation'' is given by:
 \begin{flalign*}
  \mathbb C[\![x_1,\ldots,x_k,y]\!]/(x_ix_j,(x_i-x_j)y)_{1\leq i <j\leq k} &\to&  \mathbb C[\![s,\epsilon]\!]\times \mathbb C[\![t_1]\!]\times\ldots\times \mathbb C[\![t_k]\!] &\\
  x_i &\mapsto&(\epsilon,0,\ldots,t_i,\ldots,0),i=1,\ldots,k\\
  y&\mapsto&(s,0,\ldots,0)\qquad
 \end{flalign*}
From this we compute the conductor $\mathfrak c=\langle x_1^2,\ldots,x_k^2,y\rangle$. The restriction to a tail is:
\[\omega_{C|T}\simeq\omega_T\otimes\mathcal O(2q)\simeq\mathcal O_T.\]
The restriction to the $\mathbb P^1$ underlying the ribbon is:
\[\omega_{C|R_\text{red}}\simeq \omega_{R_\text{red}}\otimes \mathcal I_{R_\text{red}/R}^\vee\otimes\mathfrak c_{R_\text{red}}^\vee\simeq\mathcal O_{R_\text{red}}(-2+(3-r)+r)\simeq\mathcal O_{R_\text{red}}(1).\]
\end{rem}
\begin{rem}
 By the same token, if we want $\omega_C$ to have non-negative restriction on every reduced curve underlying an irreducible component of the curve $C$ from Example \ref{exa:ribbonchain}, we see that $\chi(\mathcal O_{R_i})\leq 2$. In particular $\omega_C$ restricts to $\mathcal O_{\mathbb P^1}(1)$ (resp. $\mathcal O_{\mathbb P^1}$) on the component with $\chi(\mathcal O_R)=1$ (resp. $2$).
\end{rem}

\begin{dfn}
 A $(k_{1,1},\ldots,k_{1,r_1};\ldots;k_{n,1},\ldots,k_{n,r_n})$\emph{-tailed ribbon chain} is a chain of $n$ ribbons $R_i$, the $i$-th being $(k_{i,1},\ldots,k_{i,r_i})$-tailed, joined with each other as in Example \ref{exa:ribbonchain} along ``non-noded'' points. Normally, the ribbon $R_i$ has
 \[
 \chi(\mathcal O_{R_i})=
 \begin{cases} 
  r_i+2, & i=0\text{ or } n\\
  r_i+4, &\text{otherwise,}
 \end{cases}
 \]
 except for exactly one $i$, for which $\chi(\mathcal O_{R_i})$ is one lower than expected. We call such $R_i$ the \emph{special} component of the ribbon chain.
\end{dfn}
\begin{rem}
 For a tailed ribbon chain $C$, it follows once again by Noether's formula that $\omega_C$ restricts to $\mathcal O_{\mathbb P^1}$ on every reduced irreducible component of $C$, except on the special one, on which it gives $\mathcal O_{\mathbb P^1}(1)$.
\end{rem}


\begin{lem}\label{lem:ssmodelsofribbons}
 Regular semistable models of the $(k_1,\ldots,k_r)$-tailed ribbon can be classified. If the core is a smooth curve of genus two $Z$, the sets of tails $\{T_1^i,\ldots,T_{k_i}^i\}_{i=1,\ldots,r}$ belong to the same $k_i$-leaved tree, or to two conjugate ones. All the trees have the same length from root to leaves, independent of $r$ and the choice of a leaf. The configuration of attaching points on $Z$ is such that the hyperelliptic cover maps it to the corresponding configuration of noded points on $R_\text{red}$, up to reparametrisation of the latter.
\end{lem}



\begin{proof}
 A word-by-word repetition of the argument of \cite[Proposition 4.3]{B}.
\end{proof}

\begin{rem}
  If we wanted to complete the description of the semistable models of $(k_1,\ldots,k_r)$-tailed ribbons (Lemma \ref{lem:ssmodelsofribbons}) in case the core is not irreducible, the trees could be attached to any point of the core, and their length would vary accordingly.
  
 
\end{rem}

\begin{exa}
 A semistable model of the $1$-tailed ribbon $C$ (Example \ref{exa:1-tailedribbon}) can be computed by taking the pencil \[u_{+}^2f_1-tu_{-}^2p_5(f_1,f_2)=0\] on $\mathbb F_2\times\mathbb A^1_t$, with $p_5$ a generic homogeneous polynomial of degree $5$ in two variables. The resulting smoothing family is singular at six points along the central fiber, including the node of $C$. After blowing them up and normalising, we obtain a genus two curve covering the ribbon two-to-one; the tail of $C$ is attached to a Weierstrass point. This appears to be an accident due to the restriction that the smoothing comes from a pencil on $\mathbb F_2$.
\end{exa}


In order to describe semistable tails in complete generality, there is no better way than expressing the problem in terms of existence of a certain piecewise linear function on the dual graph.

\begin{prop}
 Let $C$ be a Gorenstein curve of arithmetic genus two (either an isolated singularity, or a $(k_1,\ldots,k_r)$-tailed ribbon chain). Let $\mathcal C$ be a one-parameter smoothing, and let $\phi\colon \mathcal C^{\text{ss}}\to\mathcal C$ be a semistable model. Let $Z$ be the locus on which $\phi$ is not an isomorphism. Then, up to destabilising, there exists a hyperelliptic admissible cover $\psi\colon C^{\text{ss}}\to T$, and a tropical canonical divisor $\lambda\colon\scalebox{0.8}[1.3]{$\sqsubset$}\to \mathbb R$ pulled back along $\psi$, such that $\lambda$ is supported on $Z$, and $\omega_{\mathcal C^{\text{ss}}}=\phi^*\omega_{\mathcal C}(\lambda)$.
\end{prop}
\begin{proof}
 The proof in the isolated case has appeared in \cite[\S 4.4]{B}. Up to assuming that $C$ is minimal, we may assume that the canonical linear series gives a two-to-one map $\bar{\psi}\colon \mathcal C\to \mathbb P^1$. The general fibre is the hyperelliptic cover of a smooth curve of genus two. Applying semistable reduction to the target and branch divisor, we may lift $\bar{\psi}$ to a map of nodal curves, thanks to the properness of the moduli space of admissible covers:
 \begin{center}\begin{tikzcd}
 \mathcal C^{\text{ss}}\ar[d,"\phi"]\ar[r,"\psi"] & \mathcal T\ar[d,"\phi_T"]\\
 \mathcal C\ar[r,"\bar{\psi}"] & \mathbb P^1
 \end{tikzcd}\end{center}
 Pulling back $\mathcal O_{\mathbb P^1}(1)$ to $\mathcal T$, we get a piecewise linear function on $\scalebox{0.9}[1.2]{$\top$}$ by comparing it with $\omega_{\mathcal T}(\frac{1}{2}D_B)$. Pulling back via $\mathcal C$, on the other hand, shows that:\[\phi^*\omega_{\mathcal C}=\omega_{\mathcal C^{\text{ss}}}(-\psi^*\lambda_T).\]
\end{proof}


\begin{figure}
	\centering

	
		\begin{tikzpicture} [scale=.6]
		\tikzstyle{every node}=[font=\normalsize]
		\tikzset{arrow/.style={latex-latex}}
		\def\S{2.825cm}



	
		
		\coordinate (A) at (0,6,0);
		  \coordinate (B) at (2,6,0);
		 \coordinate (C) at (3.5,5.5,0);
		 
		  \coordinate (CM) at (3.5,5,0);
		 	\coordinate (AM) at (5,6,0)
		  \coordinate (BM) at (7,6,0);
		
		 
		 \coordinate (D) at  (5,4.5,0); 
		 \coordinate (Dd) at (5.5,5.5,0);
		\coordinate (D1) at  (5,4,0)
		\coordinate (D2) at  (5,3.6,0)
		
		\coordinate (Ew) at (6.5,5,6);
		\coordinate (Ee) at (6.5,3.8,0)
		\coordinate (E) at (6.5,2,0); 
         \coordinate (E1) at (6.5,2.5,0);
         \coordinate (E2) at ( 6.5,3,0);
     
      	\coordinate (At) at (0,1,0);
		  \coordinate (Bt) at (2,1,0);
		 \coordinate (Ct) at (3.5,1,0);
		\coordinate (Dt) at  (5,1,0);
		\coordinate (Et) at (6.5,1,0); 
		\coordinate (Ddt) at (5.5,1,0);
		\coordinate (Ewt) at (6.5,1,6););
		\coordinate (BMt) at (7,1,0);
     
      
      
      	
		\coordinate (P1) at (-.3,1,1.3);
		\coordinate (P2) at (.3,1,1.7);
		\coordinate (P3) at (1.8,1,1.3);
		\coordinate (P4) at (3.3,1,1.3);
		\coordinate (P5) at (4.8,1,1.3);
		\coordinate (P6) at (6.3,1,1.3);
		\coordinate (Pp6) at (6.8,1,1.7);
		\coordinate (Q6) at (5.1,1,1.7);
        \coordinate (T6) at (6.5,1,6);
        
        \coordinate (QM6) at (6.7,1,1.3);
        \coordinate (TM6) at (7.2,1,1.6);
     
      
     
      
     
      \draw (A) to [out=130,in=50] (B);
      \draw (A) to [out=-130,in=-50] (B);
      \draw[->-] (C)--(B);
      \draw[->-] (D) to [out=70,in=70](C);
      \draw[->-] (D) to [out=-110,in=-110](C);
       \draw[->-] (E)--(D);
         
       
        \draw[->-] (Et)--(Dt);
        \draw[->-] (Dt)--(Ct);
        \draw[->-] (Ct)--(Bt);
        \draw (Bt)--(At);
                

       
		\draw[blue] (At)--(P1);
		\draw[blue] (At)--(P2);
		\draw[blue] (Bt)--(P3);
		\draw[blue] (Ct)--(P4);
		\draw[blue] (Dt)--(P5);
		\draw[blue] (Et)--(P6);
		
	
\node at (1,1,0) [above] {\small{0}};
\node at (2.75,1,0) [above] {\small{$\frac{1}{2}$}};
\node at (4.25,1,0) [above] {\small{$1$}};
\node at (5.75,1,0) [above] {\small{$\frac{3}{2}$}};

      
       
       \draw[dashed] (A)--(At);
        \draw[dashed] (B)--(Bt);
       \draw[dashed] (C)--(Ct);
       \draw[dashed] (D)--(Dt);
        \draw[dashed] (E)--(Et);


      
      \foreach \x in {A,B,C,D,At,Bt,Ct,Dt,Et}
   \fill (\x) circle (3pt);
     	\fill[red] (E) circle (3pt);



\begin{scope}[every coordinate/.style={shift={(0,7,0)}}]

       \draw ([c]A) to [out=130,in=50] ([c]B);
      \draw ([c]A) to [out=-130,in=-50] ([c]B);
       \draw[->-] ([c]C)--([c]B);
      \draw[->-] ([c]D) to [out=70,in=70]([c]C);
      \draw[->-] ([c]D) to [out=-110,in=-110]([c]C);
       \draw[->-] ([c]E1)--([c]D);
       \draw[->-] ([c]E2)--([c]D);
         
       
        \draw[->-] ([c]Et)--([c]Dt);
        \draw[->-] ([c]Dt)--([c]Ct);
        \draw[->-] ([c]Ct)--([c]Bt);
        \draw ([c]Bt)--([c]At);
                

       
		\draw[blue] ([c]At)--([c]P1);
		\draw[blue] ([c]At)--([c]P2);
		\draw[blue] ([c]Bt)--([c]P3);
		\draw[blue] ([c]Ct)--([c]P4);
		\draw[blue] ([c]Dt)--([c]P5);
		\draw[blue] ([c]Dt)--([c]Q6);
		
	
\node at (1,8,0) [above] {\small{0}};
\node at (2.75,8,0) [above] {\small{$\frac{1}{2}$}};
\node at (4.25,8,0) [above] {\small{$1$}};
\node at (5.75,8,0) [above] {\small{$2$}};

      
       
       \draw[dashed] ([c]A)--([c]At);
        \draw[dashed] ([c]B)--([c]Bt);
       \draw[dashed] ([c]C)--([c]Ct);
       \draw[dashed] ([c]D)--([c]Dt);
        \draw[dashed] ([c]E1)--([c]Et);
        \draw[dashed] ([c]E2)--([c]Et);


      
      \foreach \x in {A,B,C,D,At,Bt,Ct,Dt,Et}
   \fill ([c]\x) circle (3pt);
     	\fill[red] ([c]E1) circle (3pt);
     		\fill[red] ([c]E2) circle (3pt);
\end{scope}

 


\begin{scope}[every coordinate/.style={shift={(0,-7,0)}}]

     \draw ([c]A) to [out=130,in=50] ([c]B);
      \draw ([c]A) to [out=-130,in=-50] ([c]B);
      \draw[->-] ([c]C)--([c]B);
      \draw[->-] ([c]D1) to [out=70,in=70]([c]C);
      \draw[->-] ([c]D2) to [out=-110,in=-110]([c]C);
      \draw ([c]Ee) to [out=110,in=180]([c]D1);
      \draw ([c]Ee) to [out=-110,in=-180]([c]D2);
         
         
       
        \draw ([c]Et)--([c]Dt);
        \draw[->-] ([c]Dt)--([c]Ct);
        \draw[->-] ([c]Ct)--([c]Bt);
        \draw ([c]Bt)--([c]At);
                

       
		\draw[blue] ([c]At)--([c]P1);
		\draw[blue] ([c]At)--([c]P2);
		\draw[blue] ([c]Bt)--([c]P3);
		\draw[blue] ([c]Ct)--([c]P4);
	
		\draw[blue] ([c]Et)--([c]P6);
	    \draw[blue] ([c]Et)--([c]Pp6);
		
	
\node at (1,-6,0) [above] {\small{0}};
\node at (2.75,-6,0) [above] {\small{$\frac{1}{2}$}};
\node at (4.25,-6,0) [above] {\small{$1$}};
\node at (5.75,-6,0) [above] {\small{$0$}};

      
       
       \draw[dashed] ([c]A)--([c]At);
        \draw[dashed] ([c]B)--([c]Bt);
       \draw[dashed] ([c]C)--([c]Ct);
       \draw[dashed] ([c]D1)--([c]Dt);
        \draw[dashed] ([c]D2)--([c]Dt);
        \draw[dashed] ([c]Ee)--([c]Et);


      
      \foreach \x in {A,B,C,Ee,At,Bt,Ct,Dt,Et}
   \fill ([c]\x) circle (3pt);
     	\fill[red] ([c]D1) circle (3pt);
     		\fill[red] ([c]D2) circle (3pt);
\end{scope}


\begin{scope}[every coordinate/.style={shift={(0,-14,0)}}]

     \draw ([c]A) to [out=130,in=50] ([c]B);
      \draw ([c]A) to [out=-130,in=-50] ([c]B);
    \draw[->-] ([c]C)--([c]B);
      \draw ([c]C) to [out=130,in=50]([c]Dd);
      \draw([c]C) to [out=-130,in=-50]([c]Dd);
      \draw[->-] ([c]Ew) -- ([c]C);
     
         
         
       
        \draw [->-] ([c]Ewt)--([c]Ct);
        \draw ([c]Ddt)--([c]Ct);
        \draw[->-] ([c]Ct)--([c]Bt);
        \draw ([c]Bt)--([c]At);
                

       
		\draw[blue] ([c]At)--([c]P1);
		\draw[blue] ([c]At)--([c]P2);
		\draw[blue] ([c]Bt)--([c]P3);
	
		\draw[blue] ([c]Ddt)--([c]P5);
		\draw[blue] ([c]Ddt)--([c]Q6);
		\draw[blue] ([c]Ewt)--([c]T6);
		
		
	
\node at (1,-13,0) [above] {\small{0}};
\node at (2.75,-13,0) [above] {\small{$\frac{1}{2}$}};
\node at (4.5,-13,0) [above] {\small{$0$}};
\node at (4.9,-13,3) [left] {\small{$\frac{3}{2}$}};

      
       
       \draw[dashed] ([c]A)--([c]At);
        \draw[dashed] ([c]B)--([c]Bt);
       \draw[dashed] ([c]C)--([c]Ct);
       \draw[dashed] ([c]Dd)--([c]Ddt);
     \draw[dashed] ([c]Ew)--([c]Ewt);


      
      \foreach \x in {A,B,C,Dd,At,Bt,Ct,Ddt,Ewt}
   \fill ([c]\x) circle (3pt);
     	\fill[red] ([c]Ew) circle (3pt);
     		
\end{scope}


 

 
 \begin{scope}[every coordinate/.style={shift={(0,-23,0)}}]

       \draw ([c]A) to [out=130,in=50] ([c]B);
      \draw ([c]A) to [out=-130,in=-50] ([c]B);
       \draw[->-] ([c]CM)--([c]B);
        \draw[->-] ([c]CM)--([c]AM);
      \draw ([c]AM) to [out=130,in=50] ([c]BM);
      \draw ([c]AM) to [out=-130,in=-50] ([c]BM);
      
     
        \draw ([c]At)--([c]Bt);
        \draw[->-] ([c]Ct)--([c]Dt);
        \draw[->-] ([c]Ct)--([c]Bt);
        \draw ([c]Dt)--([c]BMt);
        
        
        
		\draw[blue] ([c]At)--([c]P1);
		\draw[blue] ([c]At)--([c]P2);
		\draw[blue] ([c]Bt)--([c]P3);
		\draw[blue] ([c]Dt)--([c]P5);
		\draw[blue] ([c]BMt)--([c]QM6);
		\draw[blue] ([c]BMt)--([c]TM6);
		
        
        \foreach \x in {A,B,AM,BM,At,Bt,Ct,Dt,BMt}
   \fill ([c]\x) circle (3pt);
     	\fill[red] ([c]CM) circle (3pt);
                
     
       
       \draw[dashed] ([c]A)--([c]At);
        \draw[dashed] ([c]B)--([c]Bt);
       \draw[dashed] ([c]CM)--([c]Ct);
       \draw[dashed] ([c]AM)--([c]Dt);
     \draw[dashed] ([c]BM)--([c]BMt);
     
     
     
\node at (1,-22,0) [above] {\small{0}};
\node at (2.75,-22,0) [above] {\small{$\frac{1}{2}$}};
\node at (4.5,-22,0) [above] {\small{$\frac{1}{2}$}};
\node at (6,-22,0) [above] {\small{$0$}};

    \end{scope}









\coordinate (Cc) at (0,0.5,0);
\coordinate (C1) at (0,1,0);
\coordinate (C2) at (0,1.5,0);
\coordinate (A1) at (1.5,3,0);

\coordinate (G) at (0.2,2.75,0);
\coordinate (F1) at (2,3,0);
\coordinate(F2) at (2,2.5,0);
\coordinate (A14) at (1.2,4,0);
\coordinate (H1) at (3.5,2.7,4);
\coordinate (H2) at (3.5,2.2,4);

\coordinate (B1) at (3.5,4.2,0);
\coordinate (B2) at (3.5,3.7,0);
\coordinate (A2) at (8,5,7);
\coordinate (A3) at (5.8,3.9,0);


\coordinate (C1t) at (0,0,0);
\coordinate (Gt) at (0.2,0,0);
\coordinate (Ft) at (2,0,0);


\coordinate (A14t) at (1.2,0,0);
\coordinate (H1t) at (3.5,0,4);


\coordinate (A1t) at (1.5,0,0);
\coordinate (B1t) at (3.5,0,0);
\coordinate (A2t) at (8,0,7);
\coordinate (A3t) at (5.8,0,0); 

\coordinate (Q1) at ( 1.2,0,1.5);
\coordinate (Q2) at ( 1.8,0,1.8);
\coordinate (Q3) at ( 8.3,0,8.9);
\coordinate (Q4) at (7.6,0,8);
\coordinate (Q5) at ( 5.2,0,1.5);
\coordinate (Q61) at ( 5.8,0,1.8);




\begin{scope}[every coordinate/.style={shift={(10,8,0)}}]


\draw[->-] ([c]C1)--([c]A1);
\draw[->-] ([c]C2)--([c]A1);
\draw[->-] ([c]A1) to [out=110,in=150] ([c]B1);
\draw[->-] ([c]A1) to [out=-70,in=-70] ([c]B2);
\draw ([c]B1) to [out=10,in=150] ([c]A3);
\draw ([c]B2) to [out=-10,in=-150] ([c]A3);
\draw ([c]B1) to [out=-40,in=110] ([c]A2);
\draw ([c]B2) to [out=-50,in=-100] ([c]A2);
     	
\draw[->-] ([c]C1t)--([c]A1t);
\draw[->-] ([c]A1t)--([c]B1t);
\draw ([c]B1t)--([c]A2t);
\draw ([c]B1t)--([c]A3t);


		\draw[blue] ([c]A1t)--([c]Q1);
		\draw[blue] ([c]A1t)--([c]Q2);
		\draw[blue] ([c]A2t)--([c]Q3);
		\draw[blue] ([c]A2t)--([c]Q4);
		\draw[blue] ([c]A3t)--([c]Q5);
		\draw[blue] ([c]A3t)--([c]Q61);

\foreach \x in {A1,A2,A3,B1,B2,C1t,A1t,B1t,A2t,A3t}
   \fill ([c]\x) circle (3pt);
 
     		
     		
     		
\node at (10.75,8,0) [above] {\small{2}};
\node at (12.5,8,0) [above] {\small{$1$}};
\node at (15.35,8,3) [right] {\small{$0$}};
\node at (14.5,8,0) [above] {\small{$0$}};



       
       \draw[dashed] ([c]C1)--([c]C1t);
        \draw[dashed] ([c]C2)--([c]C1t);
       \draw[dashed] ([c]A1)--([c]A1t);
       \draw[dashed] ([c]B1)--([c]B1t);
        \draw[dashed] ([c]B2)--([c]B1t);
        \draw[dashed] ([c]A2)--([c]A2t);
        \draw[dashed] ([c]A3)--([c]A3t);


  \fill[red] ([c]C1) circle (3pt);
    \fill[red] ([c]C2) circle (3pt);
     		

\end{scope}


\begin{scope}[every coordinate/.style={shift={(10,-1,0)}}]


\draw[->-] ([c]Cc)--([c]A1);
\draw[->-] ([c]A1) to [out=110,in=150] ([c]B1);
\draw[->-] ([c]A1) to [out=-70,in=-70] ([c]B2);

\draw ([c]B1) to [out=10,in=150] ([c]A3);
\draw ([c]B2) to [out=-10,in=-150] ([c]A3);
\draw ([c]B1) to [out=-40,in=110] ([c]A2);
\draw ([c]B2) to [out=-50,in=-100] ([c]A2);
     	
\draw[->-] ([c]C1t)--([c]A1t);
\draw[->-] ([c]A1t)--([c]B1t);
\draw ([c]B1t)--([c]A2t);
\draw ([c]B1t)--([c]A3t);


		\draw[blue] ([c]A1t)--([c]Q1);
		\draw[blue] ([c]A1t)--([c]Q2);
		\draw[blue] ([c]A2t)--([c]Q3);
		\draw[blue] ([c]A2t)--([c]Q4);
		\draw[blue] ([c]A3t)--([c]Q5);
		\draw[blue] ([c]A3t)--([c]Q61);

\foreach \x in {A1,A2,A3,B1,B2,C1t,A1t,B1t,A2t,A3t}
   \fill ([c]\x) circle (3pt);
  
     	
     		
     		
     		
     		
\node at (10.75,-1,0) [above] {\small{$\frac{3}{2}$}};
\node at (12.5,-1,0) [above] {\small{$1$}};
\node at (15.35,-1,3) [right] {\small{$0$}};
\node at (14.5,-1,0) [above] {\small{$0$}};



       
       \draw[dashed] ([c]Cc)--([c]C1t);
       
       \draw[dashed] ([c]A1)--([c]A1t);
       \draw[dashed] ([c]B1)--([c]B1t);
        \draw[dashed] ([c]B2)--([c]B1t);
        \draw[dashed] ([c]A2)--([c]A2t);
        \draw[dashed] ([c]A3)--([c]A3t);
        
         \fill[red] ([c]Cc) circle (3pt);
\end{scope}




\begin{scope}[every coordinate/.style={shift={(10,-10,0)}}]


\draw ([c]F1) to [out=0,in=70] ([c]G);
\draw ([c]F2) to [out=0,in=-70] ([c]G);
\draw[->-] ([c]F1) to  ([c]B1);
\draw[->-] ([c]F2) to  ([c]B2);
\draw ([c]B1) to [out=10,in=150] ([c]A3);
\draw ([c]B2) to [out=-10,in=-150] ([c]A3);

\draw ([c]B1) to [out=-40,in=110] ([c]A2);
\draw ([c]B2) to [out=-50,in=-100] ([c]A2);

     	
\draw ([c]Gt)--([c]Ft);
\draw[->-] ([c]Ft)--([c]B1t);
\draw ([c]B1t)--([c]A2t);
\draw ([c]B1t)--([c]A3t);


		\draw[blue] ([c]Gt)--([c]Q1);
		\draw[blue] ([c]Gt)--([c]Q2);
		\draw[blue] ([c]A2t)--([c]Q3);
		\draw[blue] ([c]A2t)--([c]Q4);
		\draw[blue] ([c]A3t)--([c]Q5);
		\draw[blue] ([c]A3t)--([c]Q61);

\foreach \x in {G,A2,A3,B1,B2,Gt,Ft,B1t,A2t,A3t}
   \fill ([c]\x) circle (3pt);
     	
     		
     		
     		
     		
\node at (11,-10,0) [above] {\small{$0$}};
\node at (13,-10,0) [above] {\small{$1$}};
\node at (15.35,-10,3) [right] {\small{$0$}};
\node at (14.5,-10,0) [above] {\small{$0$}};



       
       \draw[dashed] ([c]G)--([c]Gt);
       
       \draw[dashed] ([c]F1)--([c]Ft);
       \draw[dashed] ([c]F2)--([c]Ft);
        \draw[dashed] ([c]B2)--([c]B1t);
        \draw[dashed] ([c]A2)--([c]A2t);
        \draw[dashed] ([c]A3)--([c]A3t);
        
 
   \fill[red] ([c]F1) circle (3pt);
   \fill[red] ([c]F2) circle (3pt);
\end{scope}



\begin{scope}[every coordinate/.style={shift={(10,-19,0)}}]


\draw ([c]B1) to [out=170,in=30] ([c]A14);
\draw ([c]B2) to [out=190,in=-30] ([c]A14);

\draw[->-] ([c]H1) to ([c]B1);
\draw[->-] ([c]H2) to ([c]B2);
\draw ([c]B1) to [out=10,in=150] ([c]A3);
\draw ([c]B2) to [out=-10,in=-150] ([c]A3);

\draw ([c]B1) to [out=-40,in=110] ([c]A2);
\draw ([c]B2) to [out=-50,in=-100] ([c]A2);

     	
\draw ([c]A14t)--([c]B1t);
\draw[->-] ([c]H1t)--([c]B1t);
\draw ([c]B1t)--([c]A2t);
\draw ([c]B1t)--([c]A3t);


		\draw[blue] ([c]A14t)--([c]Q1);
		\draw[blue] ([c]A14t)--([c]Q2);
		\draw[blue] ([c]A2t)--([c]Q3);
		\draw[blue] ([c]A2t)--([c]Q4);
		\draw[blue] ([c]A3t)--([c]Q5);
		\draw[blue] ([c]A3t)--([c]Q61);

\foreach \x in {A14,A2,A3,B1,B2,H1t, A14t,B1t,A2t,A3t}
   \fill ([c]\x) circle (3pt);
  
     	
     		
     		
     		
     		
\node at (12.5,-19,0) [above] {\small{$0$}};
\node at (13.25,-19.5,2) [right] {\small{$2$}};
\node at (15.35,-19,3) [right] {\small{$0$}};
\node at (14.5,-19,0) [above] {\small{$0$}};



       
       \draw[dashed] ([c]A14)--([c]A14t);
       
       \draw[dashed] ([c]H1)--([c]H1t);
       \draw[dashed] ([c]H2)--([c]H1t);
     \draw[dashed] ([c]B1)--([c]B1t);
        \draw[dashed] ([c]B2)--([c]B1t);
        \draw[dashed] ([c]A2)--([c]A2t);
        \draw[dashed] ([c]A3)--([c]A3t);
        
 \fill[red] ([c]H1) circle (3pt);
   \fill[red] ([c]H2) circle (3pt);
\end{scope}
\end{tikzpicture}

\caption{Admissible functions on maximally degenerated tropical hyperelliptic curves: the dumbbell (l), and the theta graph (r).}
\label{fig:admredcores}
\end{figure}


For the reader's benefit, Figure \ref{fig:admredcores} provides a pictorial description of such piecewise linear functions in case the core is a genus two configuration of rational curves, including the \emph{slopes} along the nodes (or rational chains). This is where the \emph{admissible functions} of Section \ref{sec:admfun} originate from. The two columns correspond to the two maximally degenerate (stable) tropical curves of genus two. Here, the red vertex(ices) correspond(s) to the component(s) mapping to the special component(s) of the singularity/ribbon chain. Tails are not drawn, but can be understood by a tropical modification: the corresponding vertices  would lie on trees of slope $1$ towards the core, all at the same height, and lower than the red vertex(ices).

\begin{rem}
 The non-reduced structures of genus two are numerous; indeed, as opposed to the case of isolated singularities, where the $\delta$-invariant is always related to the number of branches via the genus, in the non-reduced case a very large $\delta$-invariant can always be compensated by lowering the genus of the ribbon (that can be negative). Examples with an underlying genus one curve include $\mathbb C[\![x,y]\!]/(x^2(x-y^2))$ on the tacnode, and $\mathbb C[\![x,y,z]\!]/(x^3-y^2,xz,y^2z)$ on (cusp)$\vee$(line). The ribbons should have genus $-1$ and $0$ respectively.
\end{rem}
Yet, if we require $\omega_C\geq0$, we find a characterisation of tailed ribbon chains.
\begin{prop}
 Let $C$ be a Gorenstein curve of genus two, consisting of a ribbon glued \emph{in some way} to a number of (possibly singular) rational curves. Assume that the dualising sheaf is non-negative, and positive on the ribbon. Then $C$ is a $(k_1,\ldots,k_r)$-tailed ribbon.
\end{prop}
\begin{proof}
 We may assume that the normalisation sequence is
 \[ 0 \to \mathcal O_C\to \mathcal O_R\oplus\bigoplus_{i=1}^r\mathcal O_{\mathbb P^1}^{\oplus k_i}\to\bigoplus_{i=1}^r\mathcal F_i\to 0,\]
 with $\operatorname{length}(\mathcal F_i)=\delta_i\geq k_i$; indeed, this is the minimum number of conditions that we need to impose to ensure that the values of the functions on different branches agree at the preimage of the singular point. But in fact we may even assume $\delta_i\geq k_i+1$, by the Gorenstein assumption and Lemma \ref{lem:indecomposable}.
 
 Now, it is easy to compute that for the genus of $C$ to be two we need
 \[\mathcal I_{R_\text{red}/R}\simeq \mathcal O_{\mathbb P^1}(\sum_{i=1}^r(\delta_i-k_i)-3).\]
 So, by applying Noether's formula and adjunction as in Remark \ref{rem:Noetheradjunction}, we find
 \[\omega_{C|R_{\text{red}}}\simeq\omega_{\mathbb P^1}\otimes\mathcal I_{R_\text{red}/R}^\vee\otimes\mathfrak c_{|R_{\text{red}}}^\vee\simeq \mathcal O_{\mathbb P^1}(1+\sum_{i=1}^r(k_i+\deg(\mathfrak c_{i|R_{\text{red}}}^\vee)-\delta_i)).\]
 We study the local contributions $k_i+\deg(\mathfrak c_{i|R_{\text{red}}}^\vee)-\delta_i$ around each singular point. We are then looking for Gorenstein subalgebras $A$ of $\widetilde A=\mathbb C[\![s,\epsilon]\!]\times \mathbb C[\![t_1]\!]\times\ldots\times \mathbb C[\![t_k]\!]$. By Corollary \ref{cor:principalconductor} we know that $\mathfrak c$ is principal as an ideal of $\widetilde A$; up to rescaling the generators, we may assume that
 \[\mathfrak c=\langle s^c+\epsilon s^d,t_1^{c_1},\ldots,t_k^{c_k}\rangle.\]
 If $d\geq c$, then $s^c\in \mathfrak c$; if $d<c$, then $s^{2c-d}\in\mathfrak c$. In the second case,
 \begin{align*}
  \widetilde A/\mathfrak c=&\langle 1, s,\epsilon,\ldots,\epsilon s^d(=-s^c),s^{d+1},\ldots, s^{c-1},\epsilon s^{c-1}(=s^{2c-d-1});\\
  &1,t_1,\ldots,t_1^{c_1-1};\\
  &\ldots\\
  &1,t_k,\ldots,t_k^{c_k-1}\rangle
 \end{align*}
In any case we see that $2\delta\overbrace{=}^{\text{Lemma \ref{lem:n=2delta}}}\dim_\mathbb{C}\widetilde A/\mathfrak c=2c+\sum_{i=1}^kc_i$. By Lemma \ref{lem:indecomposable} we see that $c_i\geq 2$ for all $i$. Now the condition $k+c-\delta\geq 0$ implies $\sum_{i=1}^kc_i\leq 2k$, so $c_i=2$ for all $i$.

By assumption (that the reduced subcurve underlying $R$ is smooth), we must have a generator $y$ of $A$ whose linear part contains $s+\epsilon p(s)$. Then $A$ cannot contain an element starting with $\epsilon s^a$, $a\leq c-1$, for otherwise the conductor would be larger. Then we have a basis of $A/\mathfrak c$ (which has dimension $\delta=k+c$) given by
\[(0,1,\ldots,0),\ldots,(0,0,\ldots,1),(\epsilon,0,\ldots,0),\ldots,(\epsilon s^{c-1},0,\ldots,0).\]
Up to linear algebra and automorphisms of $\widetilde A$, we may assume that the $k+1$ generators of $A$ with non-trivial linear part are of the form:
\begin{align*}
 y&=&s+\epsilon p(s)&\\
 x_i&=&\epsilon q_i(s)\oplus t_i, &\quad i=1,\ldots,k
\end{align*}
Subtracting products of $x_i$ with powers of $y$, we may even assume that $q_i$ is constant, and by Lemma \ref{lem:indecomposable} $q_i=1$. Finally, $p(s)$ can also be assumed to be constant. We recognise the singularities of Example \ref{exa:tailedribbons}.
\end{proof}



\subsection{$h^1-$vanishing}
First we describe what the core of our curves may look like.

\begin{rem}
 Up to replacing nodes with rational bridges, minimal curves of genus one are either irreducible or elliptic $m$-fold points ($m\geq 2$).
\end{rem}


\begin{defprop}
 Up to replacing nodes with rational bridges, a \emph{minimal} Gorenstein curve of genus two with isolated singularities  is one of the following:
 \begin{enumerate}
  \item irreducible;
  \item the nodal union of two minimal curves of genus one;
  \item a dumbbell configuration of $\mathbb P^1$;
  \item a theta configuration of $\mathbb P^1$;
  \item a genus one singularity with a minimal genus one branch;
  \item a type I singularity;
  \item a type II singularity.
 \end{enumerate}
\end{defprop}

\begin{lem}
 The dualising line bundle of a minimal Gorenstein curve of genus two has:
 \begin{enumerate}
 \setcounter{enumi}{4}
  \item degree $2$ on the genus one branch, or on the common branch of the two elliptic $m$-fold points;
  \item degree $2$ on the special branch;
  \item multidegree $(1,1)$ on the special branches.
 \end{enumerate}
\end{lem}


Now we describe sufficient conditions for the vanishing of higher cohomology of a line bundle on a minimal curve of genus two. This will allow us to prove the unobstructedness of the $\lambda-$aligned admissible maps satisfying the factorisation property. The proof of the following lemmas boils down to a simple albeit tedious application of the normalisation exact sequence.

\begin{lem}
 Let $C$ be a minimal Gorenstein curve of genus two with isolated singularities. A line bundle $L$ on $C$ having non-negative multidegree, positive degree on every subcurve of genus one, degree at least two, and $L\neq\omega_C$, has vanishing $h^1$.
\end{lem}

\begin{lem}
 Let $C$ be a $(k_1,\ldots,k_r)$-tailed ribbon (chain) and $L$ a non-negative line bundle on it. Then $h^1(C,L)=0$ if:
 \begin{itemize}
  \item $L$ has positive degree on at least two $k$-fold points; or
  \item $L$ restricts to $\mathcal O_{\mathbb P^1}(1)$ on $R_\text{red}$ (for the special component of the ribbon chain) and has positive degree on at least one $k$-fold point.
 \end{itemize}
\end{lem}

\section{Gorenstein curves over aligned admissible covers}\label{sec:alignedadmissiblecovers}

We work over the Artin stack of weighted admissible covers.
\begin{dfn}
 A weighted admissible cover consists of an admissible cover \[\psi\colon(C,D_R)\to(T,D_B)\] with semistable target, together with a weight function $w\colon V(\scalebox{0.8}[1.3]{$\sqsubset$})\to\mathbb N$, such that, with the induced weight function $w_T\colon V(\scalebox{0.9}[1.2]{$\top$})\to\mathbb N$ given by
 \[w_T(v)=\sum_{v^\prime\in\psi^{-1}(v)}w(v^\prime),\]
 $T$ is \emph{weighted-stable} (every weight-zero component  has at least three special points).
\end{dfn}

\subsection{The intuition and strategy} Our goal is to produce a morphism to a Gorenstein curve $\mathcal C\to\overline{\mathcal C}$ such that a line bundle on $\overline{\mathcal C}$ of degree as prescribed by the weight function $w$ would have vanishing higher cohomology. This would ensure the unobstructedness of the space of maps to projective space.

Classically, we would look for a vertical divisor $\pazocal Z$, supported on the exceptional locus of $\mathcal C\to\overline{\mathcal C}$, such that the contraction is associated to a line bundle of the form $\omega_{\mathcal C}(\pazocal Z)$ - this guarantees that $\overline{\mathcal C}$ is Gorenstein. Tropically, $\pazocal Z$ is replaced by a piecewise-linear function $\lambda$ on $\scalebox{0.8}[1.3]{$\sqsubset$}$. Since the contraction is obtained by applying the relative Proj construction to $\omega_{\mathcal C}(\lambda)$, the relevant information encoded by $\lambda$ is a collection of integral slopes along the edges of $\scalebox{0.8}[1.3]{$\sqsubset$}$. These are the objects of the stack \textbf{Div}, introduced in \cite[\S 4.2]{MarcusWise}.

Finding $\lambda$ such that $\omega_{\mathcal C}(\lambda)$ is trivial on the exceptional locus of $\mathcal C\to\overline{\mathcal C}$ reduces to a simple calculation of degrees - as opposed to a more complicated equality in the Picard group of a genus two curve - because we impose that $\lambda$ is pulled back from the admissible cover.

When the weight of the core is positive, $\lambda$ is essentially determined by the weight function. When the weight of the core is zero, instead, basically any of the positive-weight tails can be elected as the special branch of the Gorenstein singularity $\overline{C}$, be it isolated or a ribbon chain. The choice of $\lambda$ can be thought of as the choice of a degree-one divisor (i.e. a point) on the target of the admissible cover.

Given a standard tropical weighted admissible cover over $\mathbb R_{\geq 0}$, the function $\lambda$ is nonetheless uniquely determined: it is a member of the tropical canonical linear series on a subcurve $\scalebox{0.9}[1.3]{$\triangle$}\subseteq\scalebox{0.8}[1.3]{$\sqsubset$}$ (making it into a \emph{level graph} in the sense of \cite{BCGGM}) such that:
\begin{itemize}[leftmargin=.5cm]
 \item the interior $\scalebox{0.9}[1.3]{$\triangle$}^\circ$ contains every subcurve of arithmetic genus $g$ and $w\leq2g-2$;
 \item $\scalebox{0.9}[1.3]{$\triangle$}^\circ$ has weight at most $2$;
 \item $\scalebox{0.9}[1.3]{$\triangle$}$ has weight at least $3$.
\end{itemize}
At this point, we do not have a more precise recipe for finding $\lambda$ on one-parameter families. For a more general family of tropical curves, on the other hand, this determines a (polyhedral and simplicial) subdivision of the base cone. In order to prove this, we actually define $\lambda$ by interpolating among finitely many piecewise linear functions, which we call \emph{admissible}.

\subsection{Admissible functions and aligned admissible covers}\label{sec:admfun} 
Our definition of admissible functions has been inspired by that of S. Bozlee's \emph{mesa curves} \cite[\S 3]{Bozlee}. It could be said that it constitutes a Copernican revolution with respect to \cite{RSPW1}.

Let $\psi\colon(C,D_R)\to(T,D_B)$ be a weighted admissible cover over $S$, endowed with the minimal logarithmic structure. Let $\scalebox{0.9}[1.2]{$\top$}$ denote the tropicalization of the target. $\scalebox{0.9}[1.2]{$\top$}$ is therefore a weighted \emph{tree}, metrised in the monoid $\overline M_S$.

\begin{dfn}\label{def:admissiblefunction}
 Let $S$ be a geometric point. An \emph{admissible function} $\overline\lambda$ is a piecewise-linear (PL) function  on $\scalebox{0.8}[1.3]{$\sqsubset$}$ with values in $\overline M_S^\text{gp}$, defined up to a global translation by $\overline M_S^\text{gp}$, such that $K_{\scalebox{0.8}[1.3]{$\sqsubset$}}+\operatorname{div}(\overline\lambda)\geq 0$ is pulled back from a degree 1 divisor $D$ on $\scalebox{0.9}[1.2]{$\top$}$.

 
 

\end{dfn}

For the sake of concreteness, we now look at some examples, adopting the:


\begin{Convention}
With the application to stable maps in mind, we think of the source of an admissible cover as the destabilisation of a weighted-stable curve. In the following, we represent a component of positive weight by a black circle, and a component of weight zero by a white one, unless it is unstable (either a rational tail introduced as the conjugate of a rational tail of positive weight, or a rational bridge introduced by slicing $\scalebox{0.8}[1.3]{$\sqsubset$}$), in which case it is represented by a cross, and the corresponding edge is dotted. A red vertex represents the component supporting the divisor $D$, or its preimages in $\scalebox{0.8}[1.3]{$\sqsubset$}$. The blue legs (\emph{B-legs}) represent the branching divisor $D_B$ of $\psi$ (see Definition \ref{def:admissiblecover}).
\end{Convention}

\begin{exa}\label{ese:lambdabar}
Assume the core is irreducible of weight zero, and there are two tails of (large) positive weight, one of which is Weierstrass (Figure \ref{fig:exa1}).    
\begin{figure}[hbt]
\begin{tikzpicture} [scale=.4]
\tikzstyle{every node}=[font=\normalsize]
		\tikzset{arrow/.style={latex-latex}}
		
	\tikzset{cross/.style={cross out, draw, thick,
         minimum size=2*(#1-\pgflinewidth), 
         inner sep=1.2pt, outer sep=1.2pt}}
         
         
        
         
         
        
		\coordinate (Oc) at (0,3,0);
		\coordinate (T) at (-2,4,0);
		\coordinate (bT) at (-2,2,0);
		\coordinate (W) at (3,1.5,0);
	
		\coordinate (Ot) at (0,0,0);
		\coordinate (Tt) at (-2,0,0);
		\coordinate (Wt) at (3,0,0);
	
		
		\coordinate (P1) at (-.5,-1,0);
		\coordinate (P2) at (0,-1,0);
		\coordinate (P3) at (.5,-1,0);
		\coordinate (P4) at (1,-1,0);
		\coordinate (P5) at (1.5,-1,0);
    	\coordinate (P6) at (3.2,-.8,0);
		
		
			
		\begin{scope}[every coordinate/.style={shift={(0,1,0)}}]
		
	
		
	\draw  ([c]T) node [cross]{};
			
	\foreach \x in {bT,W,Tt,Wt} 
          \fill ([c]\x) circle (3pt);
     
     	
     	\draw ([c]Oc)-- ([c]bT);
     	\draw[dotted] ([c]Oc)-- ([c]T);
     	\draw ([c]Oc)-- ([c]W);
     	\draw ([c]Ot)-- ([c]Tt);
     	\draw ([c]Ot)-- ([c]Wt);
     	
     
     	\draw[blue] ([c]Ot)--([c]P1);
     	\draw[blue] ([c]Ot)--([c]P2);
     	\draw[blue] ([c]Ot)--([c]P3);
     	\draw[blue] ([c]Ot)--([c]P4);
     	\draw[blue] ([c]Ot)--([c]P5);
     	\draw[blue] ([c]Wt)--([c]P6);

       
      
       \draw[dashed] ([c]Oc)--([c]Ot);
       \draw[dashed] ([c]T)--([c]Tt);
       \draw[dashed] ([c]W)--([c]Wt);
       
      
       \draw[fill=white]  ([c]Ot) circle (3pt);
       \draw[fill=white]  ([c]Oc) node[above]{\tiny $g=2$} circle (3pt);
         \end{scope}
       \end{tikzpicture}
\caption{A weighted admissible cover with weight-zero core.}
\label{fig:exa1}
\end{figure}


There are three admissible functions $\overline{\lambda}_i$ compatible with the given weighted admissible cover; see Figure \ref{fig:exa1adm}.

 
\begin{figure}[h]
\begin{tikzpicture} [scale=.4]
\tikzstyle{every node}=[font=\normalsize]
		\tikzset{arrow/.style={latex-latex}}
		
	\tikzset{cross/.style={cross out, draw, thick,
         minimum size=2*(#1-\pgflinewidth), 
         inner sep=1.2pt, outer sep=1.2pt}}
         

\coordinate (O) at (0,0,0);
\coordinate (T1) at (-3,-3,0);
\coordinate (T1B) at (-3.8,-3,0);	
\coordinate (W) at (3,-3,0);
 
  \coordinate (W1) at (1,-3,0);
  
 
  \coordinate (T2) at (-2,-3,0);
\coordinate (T2B) at (-2.8,-3,0);	

  
 

\coordinate (s1) at (-1.5,-1.5,0);
\coordinate (s1B) at (-2.8,-2,0);	
\coordinate (sW) at (1.5,-1.5,0);	  
\coordinate (sW1) at (.5,-1.5,0);	 		  
	\coordinate (s2) at (-1,-1.5,0);
\coordinate (s2B) at (-1.8,-2,0);	

  
 
	
	

		  \begin{scope}[every coordinate/.style={shift={(-8,0,0)}}]
		  
	  
          
          \draw ([c]O)-- ([c]T1);
          \draw [dotted] ([c]O)-- ([c]T1B);
               \draw ([c]O)-- ([c]W);
        
            \node at ([c]s1) [right] {\begin{color}{blue}\scriptsize{$1$}\end{color}};   
          \node at ([c]s1B) [left] {\begin{color}{blue}\scriptsize{$1$}\end{color}}; 
           \node at ([c]sW) [right] {\begin{color}{blue}\scriptsize{$1$}\end{color}}; 
       
          \fill[red]  ([c]O) circle (3pt);
	  \draw ([c] T1B)   node [cross] {};
	  \fill ([c]W) circle (3pt);
          \fill ([c]T1) circle (3pt);
        
       
         \end{scope}
		
		
		  \begin{scope}[every coordinate/.style={shift={(2,0,0)}}]
	
        
          
          \draw ([c]O)-- ([c]T1);
          \draw [dotted] ([c]O)-- ([c]T1B);
          \draw ([c]O)-- ([c]W1);
        
          \node at ([c]s1) [right] {\begin{color}{blue}\scriptsize{$1$}\end{color}};   
          \node at ([c]s1B) [left] {\begin{color}{blue}\scriptsize{$1$}\end{color}}; 
          \node at ([c]sW1) [right] {\begin{color}{blue}\scriptsize{$3$}\end{color}}; 
       
       	  
	   \draw[fill=white]  ([c]O) circle (3pt);
	   \draw ([c] T1B)   node [cross] {};
	   \fill[red] ([c]W1) circle (3pt);
           \fill ([c]T1) circle (3pt);
       
         \end{scope}
		
 \begin{scope}[every coordinate/.style={shift={(10,0,0)}}]
          
          \draw ([c]O)-- ([c]T2);
          \draw[dotted]  ([c]O)-- ([c]T2B);
          \draw ([c]O)-- ([c]W);
        
          \node at ([c]s2) [right] {\begin{color}{blue}\scriptsize{$2$}\end{color}};   
          \node at ([c]s2B) [left] {\begin{color}{blue}\scriptsize{$2$}\end{color}}; 
          \node at ([c]sW) [right] {\begin{color}{blue}\scriptsize{$1$}\end{color}}; 
       
       
         
	    \draw[fill=white] ([c]O) circle (3pt);
	    \draw ([c] T2B)  node [cross, red] {};
  	    \fill ([c]W) circle (3pt);
            \fill[red] ([c]T2) circle (3pt);
        
        
         \end{scope}
		
		
		
	\end{tikzpicture}
	\caption{Admissible functions compatible with Figure \ref{fig:exa1}.}
	\label{fig:exa1adm}
\end{figure}

\end{exa}

\begin{rem}
 We think of $\overline\lambda$ as a collection of integral slopes on the bounded edges of $\scalebox{0.8}[1.3]{$\sqsubset$}$. It is pulled back, multiplying by the expansion factors, from a PL function $\overline\lambda_T$ on $\scalebox{0.9}[1.2]{$\top$}$ - with the \emph{caveat} that the slopes of $\overline\lambda_T$ may be half-integral on the edges of $\scalebox{0.9}[1.2]{$\top$}$ over which $\psi$ has expansion factor $2$. Indeed, it may be more accurate to say that $\overline\lambda_T$ is a PL function on the tropicalization of the orbicurve $[C/\mathfrak S_2]$. Only later, in Definition \ref{def:lambdalift}, we will fix a lift of $\overline\lambda$ to a genuine PL function with values in $\overline M_S^\text{gp}$. We shall then see that, if the core has positive weight, the number of admissible functions whose lift is non-zero is limited (compare with Example \ref{exa2:adm_fun}).
\end{rem}


\begin{rem}\label{rmk:finiteness}
 There are only \emph{finitely many} admissible functions for a given $\psi$, since there are only finitely many vertices of $\scalebox{0.9}[1.2]{$\top$}$.
\end{rem}

\begin{rem}\label{rmk:Dv}
 It follows from the Riemann-Hurwitz formula that the divisor $D$ on $\scalebox{0.9}[1.2]{$\top$}$ satisfies:
 \begin{equation}\label{eqn:Dv}
 D(v)=\operatorname{val}(v)-2+\frac{1}{2}\#\{f\ B\text{-leg }| f\multimap v\}+\sum_{e\multimap v}s(\overline\lambda_T,e),
 \end{equation}
 where $s(\overline\lambda_T,e)$ denotes the outgoing slope of $\overline\lambda_T$ along the edge $e$, and the $B$-legs represent the branching divisor $D_B$.
\end{rem}

When the core is maximally degenerate, i.e. a configuration of rational curves (and of weight zero) all the possible admissible functions are depicted in Figure \ref{fig:admredcores}. The red vertex(ices) represent(s) the one(s) supporting $K_{\scalebox{0.8}[1.3]{$\sqsubset$}}+\operatorname{div}(\overline\lambda)$. Missing from the picture is what happens outside the core, but this is easily explained: there may be any number of rational trees, on which $\overline\lambda$ has constant slope $1$ towards the core.







\begin{dfn}\label{dfn:alignmentsubdivision}
 A \emph{pre-aligned} admissible cover over a logarithmic scheme $(S,M_S)$ is one for which the values
 \begin{equation}\label{eqn:alignment}\{\overline\lambda(v)-\overline\lambda(v^\prime)|v,v^\prime\in V(\scalebox{0.8}[1.3]{$\sqsubset$})\}\subseteq (\overline M_S,\geq)\end{equation}
 are comparable, for every admissible function $\overline\lambda$ compatible with $\psi$ (Definition \ref{def:admissiblefunction}).
\end{dfn}

\begin{lem}
 The minimal logarithmic structure of a pre-aligned admissible cover is obtained, starting from the minimal logarithmic structure of the admissible cover, by adding in the elements of $\overline M_S^{\text{gp}}$ indicated in \eqref{eqn:alignment} and sharpifying.
\end{lem}

\begin{rem}
 Aligning determines a subdivision of the tropical moduli space $\sigma=\operatorname{Hom}(\overline M_S,\mathbb R_{\geq0})$, or, equivalently, a logarithmic modification of $\mathcal A_{2}^\text{wt}$, which will be denoted by $\mathcal{A}^{\text{pre}}_{2}$. See Section \ref{sec:logblowups}.
\end{rem}

 Let $\psi$ be a pre-aligned admissible cover, and $\{\overline\lambda_1,\ldots,\overline\lambda_s\}$ the set of admissible functions compatible with $\psi$. We are going to choose genuine PL functions $\lambda_i$ on $\scalebox{0.8}[1.3]{$\sqsubset$}$ with values in $\overline M_S$ lifting the $\overline\lambda_i$. In the end, we are going to define the object of interest as an interpolation/truncation of these lifts.
 
\begin{dfn}\label{def:lambdalift}
 Let $S$ be a geometric point. Define a \emph{lift} $\lambda_i\in\Gamma(S,\pi_*\overline M_C^{\text{gp}})$ of $\overline\lambda_i$ as follows. Consider the values of $\overline\lambda_i$ at the vertices of positive weight, and set to $0$ the maximal such value, unless it is attained at a unique vertex:
 \begin{itemize}[leftmargin=.5cm]
  \item of genus two, weight $w\leq 2$, and supporting $D$; or
  \item of genus one, weight $w\leq 1$; or
  \item of genus zero, weight $w\leq 2$, and supporting $D$.
 \end{itemize}
In these cases, set to $0$ the highest value of $\overline\lambda_i$ so that $w(\scalebox{0.8}[1.3]{$\sqsubset$}_{>0})\leq 2$ and $w(\scalebox{0.8}[1.3]{$\sqsubset$}_{\geq 0})\geq 3$.
\end{dfn}


\begin{exa}\label{exa2:adm_fun}
When the core has positive weight, most admissible lifts are trivial. In the examples of Figure \ref{fig:exa2}, there is only a non-trivial one (see Figure \ref{fig:exa2adm}).
\begin{figure}[h]
\begin{tikzpicture} [scale=.4]
\tikzstyle{every node}=[font=\normalsize]
		\tikzset{arrow/.style={latex-latex}}
		
	\tikzset{cross/.style={cross out, draw, thick,
         minimum size=2*(#1-\pgflinewidth), 
         inner sep=1.2pt, outer sep=1.2pt}}
         
         
        
         
         
        
		\coordinate (Oc) at (0,3,0);
		\coordinate (T1) at (-2,4,0);
		\coordinate (bT1) at (-2,2,0);
		\coordinate (T2) at (4,1.5,0);
		\coordinate (bT2) at (4,2.5,0);
		\coordinate (T3) at (5,6.5,5);
		\coordinate (bT3) at (5,7.5,5);
	
		\coordinate (Ot) at (0,0,0);
		\coordinate (T1t) at (-2,0,0);
		\coordinate (T2t) at (4,0,0);
		\coordinate (T3t) at (5,0,5);
	
		
		\coordinate (P1) at (-.5,-1,0);
		\coordinate (P2) at (0,-1,0);
		\coordinate (P3) at (.5,-1,0);
		\coordinate (P4) at (1,-1,0);
		\coordinate (P5) at (1.8,-.7,0);
    	\coordinate (P6) at (-1,-.8,0);
		
			
		\coordinate (Ot) at (0,0,0);
		\coordinate (l1t) at (-1,0.2,0);
		\coordinate (l2t) at (1.5,0,0);
		\coordinate (l3t) at (3.5,0,3);
			
		\begin{scope}[every coordinate/.style={shift={(-6,0,0)}}]
		
	
		\fill  ([c]Oc) node[above]{\tiny $g=2$}node[below]{\tiny $d=2$} circle (3pt);
			\draw  ([c]T1) node[cross]{};
			\draw  ([c]bT2) node[cross]{};
			\draw  ([c]bT3) node[cross]{};
			\fill  ([c]Ot) circle (3pt);
			\foreach \x in {bT1,T2,T3,T1t,T2t,T3t} 
          \fill ([c]\x) circle (3pt);
     
     	
     	\draw [dotted] ([c]Oc)-- ([c]T1);
     	\draw ([c]Oc)-- ([c]bT1);
     	\draw ([c]Oc)-- ([c]T2);
     	\draw [dotted]([c]Oc)-- ([c]bT2);
     	\draw ([c]Oc)-- ([c]T3);
     	\draw [dotted]([c]Oc)-- ([c]bT3);
     	
     	\draw ([c]Ot)-- ([c]T3t);
     	\draw ([c]Ot)-- ([c]T2t);
     	\draw ([c]Ot)-- ([c]T1t);
     	
     
     	\draw[blue] ([c]Ot)--([c]P1);
     	\draw[blue] ([c]Ot)--([c]P2);
     	\draw[blue] ([c]Ot)--([c]P3);
     	\draw[blue] ([c]Ot)--([c]P4);
     	\draw[blue] ([c]Ot)--([c]P5);
     	\draw[blue] ([c]Ot)--([c]P6);

    
        
           
      
       \draw[dashed] ([c]Oc)--([c]Ot);
       \draw[dashed] ([c]T1)--([c]T1t);
       \draw[dashed] ([c]bT2)--([c]T2t);
       \draw[dashed] ([c]bT3)--([c]T3t);
      
         \end{scope}
        

		\coordinate (E1) at (-1.5,3,0);
		\coordinate (E2) at (1.5,3,0);
		
		\coordinate (T1) at (-3.5,4.5,0);
		\coordinate (bT1) at (-3.5,1.5,0);
		
		\coordinate (T2) at (3.8,4.5,0);
		\coordinate (bT2) at (3.8,1.5,0);
		
	
		\coordinate (E1t) at (-1.5,0,0);
		\coordinate (E2t) at (1.5,0,0);
		\coordinate (T1t) at (-3.5,0,0);
		\coordinate (T2t) at (3.8,0,0);
	
		
		\coordinate (Pp1) at (-1.8,-1,0);
		\coordinate (Pp2) at (-1.5,-1.2,0);
		\coordinate (Pp3) at (-1.2,-1,0);
		\coordinate (Pp4) at (1.8,-1,0);
		\coordinate (Pp5) at (1.5,-1.2,0);
    	\coordinate (Pp6) at (1.2,-1,0);
		
				
	
		\coordinate (l1t) at (-2,0.,0);
		\coordinate (l2t) at (2.5,0.,0);
		\coordinate (mt) at (0,0,0);
		
		\begin{scope}[every coordinate/.style={shift={(6,0,0)}}]
		
	
			\fill  ([c]E1) node[above]{\tiny $g=1$}node[below]{\tiny $d=2$} circle (3pt);
			
			\draw  ([c]E1t) circle (3pt);
			\draw  ([c]E2t) circle (3pt);
			\draw  ([c]bT1) node[cross]{};
			\draw  ([c]bT2) node[cross]{};
			
			\foreach \x in {T1,T2,T1t,T2t} 
          \fill ([c]\x) circle (3pt);
     
     	
     	\draw ([c]E1)-- ([c]T1);
     	\draw[dotted] ([c]E1)-- ([c]bT1);
     	\draw ([c]E2)-- ([c]T2);
     	\draw[dotted] ([c]E2)-- ([c]bT2);
     	\draw ([c]E1)-- ([c]E2);
     	
     	
     	\draw ([c]E1t)-- ([c]E2t);
     	\draw ([c]E1t)-- ([c]T2t);
     	\draw ([c]E2t)-- ([c]T1t);
     	
     
     	\draw[blue] ([c]E1t)--([c]Pp1);
     	\draw[blue] ([c]E1t)--([c]Pp2);
     	\draw[blue] ([c]E1t)--([c]Pp3);
     	\draw[blue] ([c]E2t)--([c]Pp4);
     	\draw[blue] ([c]E2t)--([c]Pp5);
     	\draw[blue] ([c]E2t)--([c]Pp6);


       
               
         
       
      
       \draw[dashed] ([c]E1)--([c]E1t);
       \draw[dashed] ([c]T1)--([c]T1t);
       \draw[dashed] ([c]T2)--([c]T2t);
       \draw[dashed] ([c]E2)--([c]E2t);
      
       \draw[fill=white]  ([c]E2) node[above]{\tiny $g=1$} circle (3pt);
      
         \end{scope}
         
       \end{tikzpicture}
	\caption{Weighted admissible covers with core of positive weight.}
 \label{fig:exa2}
\end{figure}
 
         \begin{figure}[h]
\begin{tikzpicture} [scale=.45]
\tikzstyle{every node}=[font=\normalsize]
		\tikzset{arrow/.style={latex-latex}}
		
	\tikzset{cross/.style={cross out, draw, thick,
         minimum size=2*(#1-\pgflinewidth), 
         inner sep=1.2pt, outer sep=1.2pt}}
        
         \coordinate (o) at (0,0,0);
          \coordinate (t3) at (-3,-3,0);
            \coordinate (t3b) at (-3.8,-3,0);
           \coordinate (t2) at (3,-3,0);
                \coordinate (t2b) at (2,-3,0);
          \coordinate (t1) at (2,-3,-2);
              \coordinate (t1b) at (2.8,-3,-2.8);
   
         
          
         \coordinate (e1) at (-1.5,-1.5,0);
         \coordinate (v1) at (-3,-3,0);
       \coordinate (v2) at (3,-3,0);
       \coordinate (v1b) at (-3.8,-3,0);
       \coordinate (v2b) at (3.8,-3,0);
       
         
          \coordinate (s3) at (-1.5,-1.5,0);
                \coordinate (s3b) at (-2.3,-1.5,0);

        \coordinate (s2) at (1,-2.5,0);
         \coordinate (s2b) at (2.35,-2.2,0);

        
          \coordinate (s1) at (2,-1.7,-1.2);
          
       
          \coordinate (s1b) at (1.8,-1,-1.5);
          
               
                
                        

        \coordinate (ss2) at (-.5,-.5,0);
          \coordinate (ss1) at (1.5,-1.5,0);
          \coordinate (ss3) at (-2,-2,0);
            \coordinate (ss3b) at (-2.8,-2,0);
          \coordinate (ss1b) at (2.7,-2,0);

          
          
          \begin{scope}[every coordinate/.style={shift={(6,0,0)}}]
          
          
             \fill ([c]v2) circle (3pt);
              \fill ([c]v1) circle (3pt);
              \draw ([c]v1b) node [cross]{};
                  \draw ([c]v2b) node [cross]{};
            
            \draw ([c]o)--([c]e1);
            \draw ([c]o)--([c]v2);
            \draw ([c]e1)--([c]v1);
           
            \draw [dotted]([c]o)--([c]v2b);
            \draw[dotted] ([c]e1)--([c]v1b);
        
               \node at ([c]ss1) [left] {\begin{color}{blue}\scriptsize{$1$}\end{color}};   
          \node at ([c]ss2) [left] {\begin{color}{blue}\scriptsize{$1$}\end{color}}; 
             \node at ([c]ss3) [right] {\begin{color}{blue}\scriptsize{$1$}\end{color}};
         
                  \node at ([c]ss1b) [right] {\begin{color}{blue}\scriptsize{$1$}\end{color}};   
          \node at ([c]ss3b) [left] {\begin{color}{blue}\scriptsize{$1$}\end{color}}; 

	  \fill[red] ([c]e1) circle (3pt);
          \draw[fill=white] ([c]o) circle (3pt);
           
                       \end{scope}
          
          
          \begin{scope}[every coordinate/.style={shift={(-6,0,0)}}]
          
              
           \fill ([c]t1) circle (3pt);
             \fill ([c]t2) circle (3pt);
              \fill ([c]t3) circle (3pt);
               \draw ([c]t1b)node [cross]{};
             \draw ([c]t2b) node [cross]{};
              \draw ([c]t3b)node [cross]{};            

            
            \draw ([c]o)--([c]t1);
            \draw ([c]o)--([c]t2);
            \draw ([c]o)--([c]t3);
              \draw[dotted] ([c]o)--([c]t1b);
            \draw[dotted] ([c]o)--([c]t2b);
            \draw[dotted] ([c]o)--([c]t3b);
              
            
             
         \node at ([c]s1) [left] {\begin{color}{blue}\scriptsize{$1$}\end{color}};   
          \node at ([c]s2) [above] {\begin{color}{blue}\scriptsize{$1$}\end{color}}; 
           \node at ([c]s3) [right] {\begin{color}{blue}\scriptsize{$1$}\end{color}}; 
             \node at ([c]s1b) [left] {\begin{color}{blue}\scriptsize{$1$}\end{color}};   
          \node at ([c]s2b) [left] {\begin{color}{blue}\scriptsize{$1$}\end{color}}; 
           \node at ([c]s3b) [left] {\begin{color}{blue}\scriptsize{$1$}\end{color}}; 
           
       \fill[red] ([c]o) circle (3pt);
           
           
                       \end{scope}

\end{tikzpicture}
\caption{Admissible functions compatible with Figure \ref{fig:exa2}.}
\label{fig:exa2adm}
\end{figure}
\end{exa}


\begin{rem}
When $S$ is a geometric point, there is no obstruction to finding the required lifts $\lambda_i$.
\end{rem}

\begin{defprop}
Define a PL function on (a subdivision of) $\scalebox{0.8}[1.3]{$\sqsubset$}$ by:
 \[\lambda=\max\{0,\lambda_1,\ldots,\lambda_s\}.\]
\end{defprop}
\begin{proof}

For $\lambda$ to be well-defined, we need to argue that, for every vertex $v$ of $\scalebox{0.8}[1.3]{$\sqsubset$}$, the values $\{\lambda_i(v)|i=1,\ldots,s\}$ are comparable, and there is a unique way of subdividing $\scalebox{0.8}[1.3]{$\sqsubset$}$ in order to make $\lambda$ piecewise-linear.

We give an argument for the case of weight-zero core (the case of positive weight is easier). It is convenient to work on $\scalebox{0.9}[1.2]{$\top$}$. The leaves of $\scalebox{0.9}[1.2]{$\top$}$ have positive weight, and the value of $\lambda_T$ on them is determined: it usually is $0$, unless a leaf $v$ has weight $w(v)\leq 2$, in which case there may be a unique $i$ for which $\lambda_i(v)>0$ ($\lambda_i$ is the unique admissible function such that $K_{\scalebox{0.8}[1.3]{$\sqsubset$}}+\operatorname{div}(\lambda_i)$ is supported on $v$).

So, in order to prove that the maximum is well defined at every vertex, we can proceed inductively from the leaves and run through all of $\scalebox{0.9}[1.2]{$\top$}$. Assume that $\lambda_T(v_1)$ has been determined, then we need to establish the behaviour of $\lambda_T$ on the edge $e$ between $v_1$ and $v_2$, and its value on $v_2.$ Upon relabelling, we may suppose that $\lambda_1$ is the function with maximal slope along $e$ among the ones with $\lambda_i(v_1)=\lambda_T(v_1)$; so $\lambda_T$ coincides with $\lambda_1$ in a neighbourhood of $v_1$. 
If  $\lambda_1$ is also the function of maximal slope along $e$ among all the admissible functions, then $\lambda_T(v_2)=\lambda_1(v_2)$ and we are done.

If not, there must be a function $\lambda_2$ with slope along $e$ greater than that of $\lambda_1$ but $\lambda_2(v_1)\leq\lambda_1(v_1).$
First, assume that $\overline\lambda_1$ and $\overline\lambda_2$ differ only along $e$, i.e. $D(\lambda_1,v_2)=1$ and $D(\lambda_2,v_1)=1$. Then there is a unique way to interpolate between $\lambda_1$ and $\lambda_2$ along $e$, namely by solving the equations:
 \[
 \left\{
 \begin{array}{ r @{{}={}} l }
  \ell(e)&\ell(e^\prime)+\ell(e^{\prime\prime})\\
  \lambda_2(v_2)-\lambda_1(v_1) & s(\lambda_1,e)\ell(e^\prime)+s(\lambda_2,e)\ell(e^{\prime\prime}).
 \end{array}
 \right.
\]
This uniquely determines the subdivision of $e$ on which $\lambda_T$ is piecewise-linear, and the values of $\lambda_T$ on both $v_2$ and the newly introduced vertex $v_{1,2}.$

More generally, $\lambda_1$ and $\lambda_2$ will differ along the chain $R$ connecting the support of $D_1$ to that of $D_2$, and containing the edge $e$.
 Then $\lambda_2-\lambda_1=D_2-D_1$ is a rational function with constant slope $1$ along $R$.
 In order to define $\lambda_T$ we need to interpolate between $\lambda_1$ and $\lambda_2$ along $R$, see Figure~\ref{fig:lambdacomparison}.
 
 
\begin{figure}[htb]
	\centering


		\begin{tikzpicture} [scale=.6]
		\tikzstyle{every node}=[font=\normalsize]
		\tikzset{arrow/.style={latex-latex}}

\draw[name path=E] ellipse (5.8 and 3.6);
\draw[dashed,name path=L1] (3.5,3) to (3,-3.2);
\draw[dashed,name path=L2] (-3,3.2) to (-3,-3.3);

\begin{scope}[on background layer]
   
\filldraw[draw=black,bottom color=blue, top color=white]
    [intersection segments={of= L2 and E}];
    
   
    \filldraw[bottom color=white, top color=green]
   [intersection segments={of=E and L1}];
    \path[name path=line1] (3,-3.1) --  (5.7,0);
    \fill[bottom color=white, top color=green]
[intersection segments={of=line1 and E}];
 \end{scope}
 \coordinate (A) at (-4.5,-0.2);
  \coordinate (B) at (-3,1);
  \coordinate (C) at (-1,-.2);

    \coordinate (C1) at (.5,-.7);
    \coordinate (D) at (1.9,0.5);
    \coordinate(E) at (3.2,-0.8);
    \coordinate (F) at (4.8,-.2);

\draw[->-] (A) to (B);
\draw[->-] (B) to (C);
\draw[dotted] (C) to (C1);

\draw[->-] (C1) to (D);
\draw[->-] (D) to (E);
\draw[->-] (E) to (F);
\draw[thick] (B) to (-3,3.2);
\draw[thick] (E) to (3,-3.2);

\node at (-3,3.6) [above]  {\small{\begin{color}{green}$\lambda_2$\end{color}<\begin{color}{blue}$\lambda_1$\end{color}=0}};
\node at (-3,3.1) [above]  {\small{$\infty_1$}};

\node at (-4.5,-0.3) [below] {\color{red}\small{$D_2$}};
\node at (-4,0.3) [above] {\small{$R_2$}};
\node at (-2.6,1) [above] {\small{$v_1$}};
\node at (-1.7,0.2) [above] {\small{$e_1$}};
\node at (0,-1.3) [below] {R};
\node at (1.9,.5) [above left] {\small{$v'$}};
\node at (1.9,.5) [above right] {\color{red}\small{$D'_1$}};
\node at (2.7,-.4) [left] {\small{$e_k$}};
\node at (2.8,-.9) [below] {\small{$v_2$}};
 \node at (4,-.6) [below] {\small{$R_1$}};
 
 \node at (3,-3.6) [below]  {\small{\begin{color}{blue}$\lambda_1$\end{color}<\begin{color}{green}$\lambda_2$\end{color}=0}};
\node at (3,-3.1) [below]  {\small{$\infty_2$}};
 \node at (4.6,-.2) [above] {\color{red}\small{$D_1$}};

 \foreach \x in {B,E}
	\fill (\x) circle (3pt);
\foreach \x in {A,D,F}
	\fill[red] (\x) circle (3pt);
\end{tikzpicture}
\caption{Comparison of $\lambda_1$ and $\lambda_2$.}
\label{fig:lambdacomparison}
\end{figure}



We will argue that this is possible because we can always inductively reduce to the above case, for which we have an explicit formula for the interpolation.
 
  We may assume that $R_2$ has zero length, because changing $\lambda_2$ on $R_2$ increases it without affecting $\lambda_2\leq\lambda_1$ (in the blue-shaded region), and similarly for $R_1$.
  
  Now let $\overline\lambda_1^\prime$ be the function that differs from $\overline\lambda_1$ only on the last edge $e_k$ of $R$. If $\lambda_1^\prime(\infty_1)=0$, then $\lambda_1^\prime\geq\lambda_1$, so that we can instead compare $\lambda_1^\prime$ and $\lambda_2$, which is possible by the inductive assumption. If instead $\lambda_1^\prime(\infty_1)<0$, then $\lambda_1^\prime(\infty_1^\prime)=0$ for some $\infty_1^\prime$ in the green-shaded region (since $\lambda_1^\prime<\lambda_1$ on the complement). In fact, it is easy to see that $\lambda_1^\prime(\infty_2)=0$, so $\lambda_2(v^\prime)=\lambda_1^\prime(v^\prime)<\lambda_1(v^\prime)$, and we conclude that $\lambda_T$ interpolates between $\lambda_1$ and $\lambda_2$ (equivalently $\lambda_1^\prime$) only along $e_k$ - which we have already solved.
 \end{proof}
 
\begin{dfn}\label{dfn:furthersubdivision}
The bending locus of $\lambda$ determines a subdivision $\widetilde{\scalebox{0.8}[1.3]{$\sqsubset$}}$ of $\scalebox{0.8}[1.3]{$\sqsubset$}$. This in turn determines a further subdivision of $\operatorname{trop}(\mathcal A_2^{\text{pre}})$.
\end{dfn}

\begin{lem}
  $\lambda$ is stable under specialisation, so it defines a section $\lambda\in\Gamma(S,\pi_*\overline{M}_{\widetilde C}^{\text{gp}})$.
 \end{lem}
 \begin{proof}
  Let $t\rightsquigarrow s$ be a specialisation of geometric points in $S$, corresponding to an edge contraction $\scalebox{0.8}[1.3]{$\sqsubset$}_s\to\scalebox{0.8}[1.3]{$\sqsubset$}_t$. Then, for any admissible lift $\lambda_{i,t}$ on $\scalebox{0.8}[1.3]{$\sqsubset$}_t$, there exists an admissible lift $\lambda_{j(i),s}$ specialising to it.
 \end{proof}



Given a cone $\sigma$ of $\operatorname{trop}(\mathcal A_{2}^\text{wt})$, let us denote by $\Sigma$ the subdivision of $\sigma$ induced by aligning (Definition \ref{dfn:alignmentsubdivision}) and linearising (Definition \ref{dfn:furthersubdivision}) $\lambda$. Once we cut $\lambda_i$ at $0$, comparing its values at certain vertices may become obsolete. Therefore, we introduce a coarsening $\Sigma^\prime$ of $\Sigma$, with the advantage of being simplicial.

\begin{dfn}
 Let $\scalebox{0.8}[1.3]{$\sqsubset$}$ be a tropical curve metrised in $\mathbb R_{>0}$, and $\lambda\colon\scalebox{0.8}[1.3]{$\sqsubset$}\to\mathbb R_{\geq0}$ a PL function. The \emph{combinatorial type} $[\lambda]$ of $\lambda$ encodes the subdivision of $\mathbb R_{\geq 0}$ induced by the image of the vertices of $\scalebox{0.8}[1.3]{$\sqsubset$}$, and the correspondence between the edges of $\scalebox{0.8}[1.3]{$\sqsubset$}$ and those of $\mathbb R_{\geq 0}$ together with the slopes, forgetting all the metric information.
\end{dfn}


\begin{dfn}\label{dfn:coarseningsubdivision}
 Let $\Sigma^\prime$ be the coarsening of $\Sigma$ obtained by joining those cones on which $\lambda$ has constant combinatorial type: \[\Sigma^\prime=\{\tau^\prime\},\qquad\text{where}\qquad \tau^\prime=\bigcup_{\tau\in\Sigma|[\lambda_{|\tau}]\text{ is fixed}}\tau.\]
\end{dfn}

\begin{defprop}\label{defprop:Atilde}
 There is a logarithmically \'{e}tale model $a\colon\widetilde{\mathcal A}_{2}\to\mathcal{A}^{\text{wt}}_{2}$ on which $\widetilde{\scalebox{0.8}[1.3]{$\sqsubset$}}$ is the tropicalization of a family of genus two curves $\widetilde{\mathcal C}$ partially destabilising $a^*\mathcal{C}$, $\lambda\in\Gamma(\widetilde{\mathcal C},\overline M_{\widetilde{\mathcal C}})$, and all of its values at vertices of $\widetilde{\scalebox{0.8}[1.3]{$\sqsubset$}}$ are comparable.
\end{defprop}
\begin{proof}
 The subdivision $\Sigma^\prime$ of Definition \ref{dfn:coarseningsubdivision} coincides with the one obtained from the following procedure. If $\sigma$ denotes (a cone of) the tropicalization of $\mathcal{A}^{\text{wt}}_{2}$, with universal curve $\scalebox{0.8}[1.3]{$\sqsubset$}$ and a PL map $\lambda\colon\scalebox{0.8}[1.3]{$\sqsubset$}\to\mathbb{R}_{\geq0}\times\sigma$, first subdivide $\scalebox{0.8}[1.3]{$\sqsubset$}$ along the bending locus of $\lambda$, then subdivide the target $\mathbb{R}_{\geq0}\times\sigma$ along the image of cones of $\widetilde\scalebox{0.8}[1.3]{$\sqsubset$}$ under $\lambda$. This in turn determines a further subdivision of $\widetilde\scalebox{0.8}[1.3]{$\sqsubset$}$, and one of $\sigma$ such that:
 \begin{enumerate}[leftmargin=.5cm]
  \item \emph{equidimensionality}: every cone of $\widetilde\scalebox{0.8}[1.3]{$\sqsubset$}$ surjects onto one of $\Sigma^\prime$;
  \item \emph{reducedness}: for every cone of $\widetilde\scalebox{0.8}[1.3]{$\sqsubset$}$, the projection is surjective at the level of lattices.
 \end{enumerate}
Importantly, the process stops. See \cite[\S 2]{DhruvExpansions} (particularly $\S2.8$) and references therein for more details. These requirements ensure that $\tilde\pi\colon\widetilde{\mathcal C}\to \widetilde{\mathcal A}_2$ is a flat family of prestable curves. The model $\widetilde{\mathcal A}_{2}$ can be constructed by pulling back a modification of the Artin fan \cite{ACMUW}.

Note that, due to the existence of some simplicial (but not smooth) cones in the subdivision, in order for the lattice map to be surjective it may be necessary to perform a Kummer extension of the base logarithmic structure \cite[\S 4]{BV}:
\[M_{\mathcal A}\subseteq M_{\widetilde{\mathcal A}}\subseteq\frac{1}{2}M_{\mathcal A}^\text{gp},\]
as prescribed by the subdivision $\widetilde{\scalebox{0.8}[1.3]{$\sqsubset$}}$ and the slopes of $\lambda$, see Figure \ref{fig:halfintegers}. This operation should be thought of as a generalised root stack construction, or generalising the presentation of a simplicial affine toric variety as the coarse moduli of $[\mathbb A^n/G]$, $G$ the finite abelian group encoding the difference between the two lattices at stake.
\end{proof}

\begin{figure}[htb]
	\centering

	
		\begin{tikzpicture} [scale=.6]
		\tikzstyle{every node}=[font=\normalsize]
		\tikzset{arrow/.style={latex-latex}}
		
		\tikzset{cross/.style={cross out, draw, thick,
         minimum size=2*(#1-\pgflinewidth), 
         inner sep=1.5pt, outer sep=1.5pt}}
\coordinate (A) at (0,2);
\coordinate (B) at (-2,1);
\coordinate (C) at (2,5);

\coordinate (B1) at (-2,3.5);
\coordinate (C1) at (2.8,4); 

\coordinate (At) at (0,-1);
\coordinate (Bt) at (-2,-1);
\coordinate (Ct) at (2,-1);
\coordinate (C1t) at (2.8,-1); 

 \foreach \x in {B,C,Bt,Ct}
	\fill (\x) circle (3pt);

\draw (A)--(B);
\draw (A)--(C);
\draw (At)--(Bt);
\draw (At)--(Ct);
\node at (B) [left] {\scriptsize{$l_1$}};
\node at (-1,1.5) [above]  {\color{blue}\small{$1$}};
\node at (C) [right] {\scriptsize{$l_2$}};
\node at (0.8,3.3) [above]  {\color{blue}\small{$3$}};
\node at (0.1,2) [below right] {\scriptsize{$\frac{1}{2}(3e+3l_1-l_2)$}};


\draw [decorate,decoration={brace,amplitude=5pt,mirror},xshift=0.4pt,yshift=-0.4pt] ([c]Bt)--([c]At) node[black,midway,xshift=-0.4cm,yshift=-0.4cm] {\scriptsize{$\frac{1}{2}(3e+l_1+l_2)$}};
\draw [thick,decorate,decoration={brace,amplitude=5pt},xshift=0.4pt,yshift=-0.4pt] ([c]Bt)--([c]Ct) node[black,midway,yshift=0.4cm] {\scriptsize{$e$}};
\draw [thick,decorate,decoration={brace,amplitude=5pt,mirror},xshift=0.4pt,yshift=-0.4pt] ([c]At)--([c]Ct) node[black,midway,xshift=0.6cm,yshift=-0.4cm] {\scriptsize{$\frac{1}{2}(l_2-l_1-e)$}};

\draw[red,fill=red] ([c]At) circle (3pt);
\draw[red,fill=red] ([c]A) circle (3pt);




\begin{scope}[every coordinate/.style={shift={(10,0)}}]
\foreach \x in {B1,C1,Bt,C1t}
	\fill ([c]\x) circle (3pt);

\draw ([c]A)--([c]B1);
\draw ([c]A)--([c]C1);
\draw ([c]At)--([c]Bt);
\draw ([c]At)--([c]C1t);
\node at ([c]B1) [left] {\scriptsize{$m_1$}};
\node at (9,2.8) [above]  {\color{blue}\small{$1$}};
\node at ([c]C1) [right] {\scriptsize{$m_2$}};
\node at (11,3) [above]  {\color{blue}\small{$1$}};
\node at ([c]A) [below right] {\scriptsize{$\frac{1}{2}(m_1+m_2-f)$}};
\draw [decorate,decoration={brace,amplitude=5pt,mirror},xshift=0.4pt,yshift=-0.4pt] ([c]Bt)--([c]At) node[black,midway,xshift=-0.4cm,yshift=-0.4cm] {\scriptsize{$\frac{1}{2}(f+m_1-m_2)$}};
\draw [thick,decorate,decoration={brace,amplitude=5pt},xshift=0.4pt,yshift=-0.4pt] ([c]Bt)--([c]C1t) node[black,midway,yshift=0.4cm] {\scriptsize{$f$}};
\draw [thick,decorate,decoration={brace,amplitude=5pt,mirror},xshift=0.4pt,yshift=-0.4pt] ([c]At)--([c]C1t) node[black,midway,xshift=0.6cm,yshift=-0.4cm] {\scriptsize{$\frac{1}{2}(f+m_2-m_1)$}};

\draw[red,fill=red] ([c]At) circle (3pt);
\draw[red,fill=red] ([c]A) circle (3pt);

\end{scope}
\end{tikzpicture}
\caption{The two situations in which half-integral lengths occur.}
\label{fig:halfintegers}
\end{figure}


\begin{lem}
 The minimal logarithmic structure of $\widetilde{\mathcal A}_{2}$ is locally free.
\end{lem}
\begin{proof}
 The construction of the minimal logarithmic structure can be traced back to Definition-Proposition \ref{defprop:Atilde}. The fact that it is locally free can be explained as follows: $\lambda$ provides us with a map from (a subdivision of) $\widetilde{\scalebox{0.8}[1.3]{$\sqsubset$}}$ to (a subdivision of) $\mathbb R_{\geq 0}$. The rank of the logarithmic structure is the number of levels (finite edges in the polyhedral subdivision) of $\mathbb R_{\geq 0}$. For every level, there is at most one edge of $\widetilde{\scalebox{0.8}[1.3]{$\sqsubset$}}$ mapping to it with expansion factor $>1$: the corresponding smoothing parameter can be taken as a generator of the base logarithmic structure.
\end{proof}

Summing up, we have proven the following:
\begin{thm}\label{thm:tildeA}
 The moduli space $\widetilde{\mathcal A}_{2}$ of aligned admissible covers is a logarithmically smooth DM stack with locally free logarithmic structure (and therefore smooth) admitting a log \'{e}tale morphism to ${\mathcal A}_{2}^\text{wt}$.
\end{thm}
\begin{proof}
 Log smoothness follows, since $\widetilde{\mathcal A}_{2}$ has been constructed by pulling back a log \'{e}tale modification of the Artin fan of $\mathcal A_{2}^\text{wt}$.
\end{proof}

We extract the combinatorial properties of $\lambda$ that we are going to use in order to prove the unobstructedness of the moduli space of aligned admissible maps.

 \begin{dfn}
  We shall denote by $\scalebox{0.9}[1.3]{$\triangle$}\subseteq\widetilde{\scalebox{0.8}[1.3]{$\sqsubset$}}$ the closure of the support of $\lambda$.
 \end{dfn}
 
\begin{lem}
There is always at least one vertex of positive weight on $\partial\scalebox{0.9}[1.3]{$\triangle$}$.
\end{lem}


\begin{lem}\label{lem:uniquevertexonboundary}
 Assume $w(\scalebox{0.9}[1.3]{$\triangle$}^\circ)=0$.  If there is only one vertex $v$ of positive weight on $\partial\scalebox{0.9}[1.3]{$\triangle$}$, then $D(v)=1$ and $w(v)\geq 3$.
\end{lem}
\begin{proof}
 If $v$ were not special, it would be possible to ``push'' $D$ towards $v$ until another vertex of positive weight reaches $\partial\scalebox{0.9}[1.3]{$\triangle$}$; pushing $D$ makes $\lambda$ increase, which is a contradiction. The second claim follows immediately from the definitions.
\end{proof}

Similarly, the following observations can be made.

\begin{lem}
 If the vertex $v$ supporting $D$ is in $\scalebox{0.9}[1.3]{$\triangle$}^\circ_T$ and $w(v)=0$, then it disconnects $\scalebox{0.9}[1.3]{$\triangle$}_T$ and there is at least one vertex of positive weight on $\partial\scalebox{0.9}[1.3]{$\triangle$}_T$ on either side of $v$.
\end{lem}

\begin{lem}\label{lem:highergenusonbdry}
 A vertex of $\partial\scalebox{0.9}[1.3]{$\triangle$}$ can have positive genus only if it supports $D$, it has genus one, and so does $\scalebox{0.9}[1.3]{$\triangle$}^\circ$.
\end{lem}




\subsection{Examples of subdivision}
In this section we collect a few examples to show how the subdivisions  $\Sigma$ and $\Sigma^\prime$ of a cone $\sigma\in\operatorname{trop}(\mathcal{A}_2^{\text{wt}})$,  the combinatorics of $\lambda$ on the various cones, and the associated singularities in $\overline{\mathcal C}$ look like. The construction of $\overline{\mathcal C}$ will be carried out in the next section.

\begin{exa}
 The stabilisation of $C$ has smooth core of weight zero, and two rational tails of high weight, one of which attached  to a Weierstrass point (compare with Example \ref{ese:lambdabar}).
In this case, the subdivisions $\Sigma$ and $\Sigma^\prime$ coincide. Notice that there is a simplicial non-smooth cone $\sigma_3$; correspondingly, half edge-lengths occur in a Kummer extension of $M_{\widetilde{A}}$. See Figure \ref{esesub1}.  
\begin{figure}[h]
\begin{tikzpicture} [scale=.4]
\tikzstyle{every node}=[font=\normalsize]
		\tikzset{arrow/.style={latex-latex}}
		
	\tikzset{cross/.style={cross out, draw, thick,
         minimum size=2*(#1-\pgflinewidth), 
         inner sep=1.2pt, outer sep=1.2pt}}
         
         
        
         
         
        
		\coordinate (Oc) at (0,3,0);
		\coordinate (T) at (-2,4,0);
		\coordinate (bT) at (-2,2,0);
		\coordinate (W) at (3,1.5,0);
	
		\coordinate (Ot) at (0,0,0);
		\coordinate (Tt) at (-2,0,0);
		\coordinate (Wt) at (3,0,0);
	
		
		\coordinate (P1) at (-.5,-1,0);
		\coordinate (P2) at (0,-1,0);
		\coordinate (P3) at (.5,-1,0);
		\coordinate (P4) at (1,-1,0);
		\coordinate (P5) at (1.5,-1,0);
    	\coordinate (P6) at (3.2,-.8,0);
		
		
			
		\begin{scope}[every coordinate/.style={shift={(2,5,0)}}]
		
		
     
     	
     	\draw[dotted] ([c]Oc)-- ([c]T);
     	\draw ([c]Oc)-- ([c]bT);
     	\draw ([c]Oc)-- ([c]W);
     	\draw ([c]Ot)-- ([c]Tt);
     	\draw ([c]Ot)-- ([c]Wt);
     	
     
     	\draw[blue] ([c]Ot)--([c]P1);
     	\draw[blue] ([c]Ot)--([c]P2);
     	\draw[blue] ([c]Ot)--([c]P3);
     	\draw[blue] ([c]Ot)--([c]P4);
     	\draw[blue] ([c]Ot)--([c]P5);
     	\draw[blue] ([c]Wt)--([c]P6);

      
       
       \node at (1,8.5,0) [above] {\scriptsize{$l_1$}};
        \node at (.9,8.4,0) [below] {\scriptsize{$l_1$}};
        \node at (3.5,7.2,0) [above] {\scriptsize{$l_2$}};
        
       \node at (1,5,0) [above] {\scriptsize{$l_1$}};
        \node at (3.5,5,0) [above] {\scriptsize{$2l_2$}};
        
       
      \node at (0,9,0) [left] {\scriptsize{$\overline{T}$}};
        \node at (0,7,0) [left] {\scriptsize{$T$}};
         \node at (5,6.2,0) [right] {\scriptsize{$W$}};
        
       
      
       \draw[dashed] ([c]Oc)--([c]Ot);
       \draw[dashed] ([c]T)--([c]Tt);
       \draw[dashed] ([c]W)--([c]Wt);
       
      
		\draw[fill=white]  ([c]Oc) circle (3pt);
		\draw  ([c]T) node [cross]{};
		\draw[fill=white]  ([c]Ot) circle (3pt);
	   \foreach \x in {bT,W,Tt,Wt} 
          \fill ([c]\x) circle (3pt);
       
         \end{scope}

		\coordinate (O) at (0,0,0);
		\coordinate (l2) at (6,0,0);
		\coordinate (l1) at (0,6,0);
		\coordinate (r1) at (3,6,0);
		\coordinate (r2) at (5,5,0);
		\coordinate (r3) at (6,2,0);
		
		\begin{scope}[every coordinate/.style={shift={(0,-8,0)}}]
		
	
		\draw[->] ([c]O)--([c]l2);
		\draw[->] ([c]O)--([c]l1);
		\draw[green] ([c]O)--([c]r1);
		\draw[orange]([c]O)--([c]r2);
		\draw[red] ([c]O)--([c]r3);
		
		 \node at (.7,-4.5,0) [above] {\scriptsize{$\sigma_1$}};
		 \node at (3,-2,0) [above] {\begin{color}{green}\scriptsize{$\tau_1$}\end{color}\scriptsize{, $l_2=2l_1$}};
		
		 \node at (2.3,-5,0) [above] {\scriptsize{$\sigma_2$}};
		 \node at (5,-3,0) [above right] {\begin{color}{orange}\scriptsize{$\tau_2$}\end{color}\scriptsize{, $l_2=l_1$}};
		
		 \node at (3.4,-6.2,0) [above] {\scriptsize{$\sigma_3$}};
		 \node at (6,-6,0) [above right] {\begin{color}{red}\scriptsize{$\tau_3$}\end{color}\scriptsize{, $l_1=3l_2$}};
		
		 \node at (3.8,-7.7,0) [above] {\scriptsize{$\sigma_4$}};
		 \node at ([c]l2) [right] {\scriptsize{$l_1$}};
		  \node at ([c]l1) [left] {\scriptsize{$l_2$}};
		  \end{scope}
		 
		  
		  
		 
		   \coordinate (O2) at (-.5,.5,0);
		   \coordinate (B1) at (-.7,-1.5,0);
		   \coordinate (C1) at (-2.9,-1.5,0);
		  \coordinate (B2) at (1,-1.5,0);
		  \coordinate (B2l) at (2.3,-1.5,0);
		  \coordinate(C2) at (3.5,-1.5,0);
		 
		  \coordinate (O22) at (-.5,.8,0);
		  \coordinate (B12) at (-.9,-.8,0);
		  \coordinate (C12) at (-2.3,-1.5,0);
		  
		  
		   \coordinate (O3) at (0,.9,0);
		   \coordinate (D2) at (.5,-.7,0);
		  
		  
		  	 
		  \coordinate (Bt1) at (-1,-1.5,0);
	     
	      
	       
	          \coordinate (D22) at (.5,-1.5,0);
	      
	     \coordinate (y1) at (6,0,0);
	   \coordinate (y2) at (6,-3,0);
	    \coordinate (x1) at (4.5,-1.5,0);
	   \coordinate (x2) at (7.5,-1.5,0);
	   \coordinate (oo) at (6,-1.5,0);
	    \coordinate (m1) at (7,-.5,0);
	   \coordinate (m2) at (5,-.5,0);
	    \coordinate (b1) at (5.7,-.4,0);
	    \coordinate (b2) at (7,.2,0);
	    
	     
	       \coordinate (y1e) at (6.2,0,0);
	   \coordinate (y2e) at (6.2,-3,0);
	   \coordinate (b1e) at (5.7,-2.7,0);
	    \coordinate (b2e) at (7,-2.2,0);
	     
	       \coordinate (m3) at (7,-2.5,0);
		  
		 
		  \begin{scope}[every coordinate/.style={shift={(15,12,0)}}]
		  
		  \node at (12,12,0) [left] {{$\sigma_1:$}};
		  
		
          
          \draw ([c]O2)-- ([c]B2);
          \draw ([c]O2)-- ([c]B1);
          \draw ([c]B1)-- ([c]C1);
          \draw ([c]B2)-- ([c]B2l);
          \draw ([c]B2l)-- ([c]C2);
          
         
          \node at (14.5,11.7,0) [left] {{\begin{color}{blue}\scriptsize{$2$}\end{color}}};
           \node at (15.3,11.7,0) [right] {{\begin{color}{blue}\scriptsize{$1$}\end{color}}};
          
         
          \draw [decorate,decoration={brace,amplitude=5pt,mirror},xshift=0.4pt,yshift=-0.4pt] ([c]O2)--([c]B2) node[black,midway,xshift=-0.1cm,yshift=-0.8cm] {\scriptsize{$2l_1$}};
           \draw [decorate,decoration={brace,amplitude=5pt,mirror},xshift=0.4pt,yshift=-0.4pt] ([c]B2)--([c]B2l) node[black,midway,xshift=0.2cm,yshift=-0.4cm] {\scriptsize{$l_2-2l_1$}};
           
          
           \draw[gray] ([c]y1)-- ([c]y2);
            \draw[thick] ([c]x1)-- ([c]x2);
              \draw[thick] ([c]b1)-- ([c]b2);
              \draw[gray] ([c]oo) parabola ([c]m1);
               \draw[gray] ([c]oo) parabola ([c]m2);
            
           
            \node at ([c]m1) [right] {{\scriptsize{$\overline{T}$}}};
             \node at ([c]x2) [right] {{\scriptsize{$T$}}};
              \node at ([c]b2) [right] {{\scriptsize{$W$}}};
              
            
                
		  \draw[fill=white]  ([c]O2) circle (3pt);
		  
		
          \fill ([c]B2l) circle (3pt);
            \draw[fill=red]([c]B1) circle (3pt);
            
          \draw ([c]B2) node[cross]{};
              
            \end{scope}
	
	

		  \begin{scope}[every coordinate/.style={shift={(15,8,0)}}]
		  
		  \node at (12,8,0) [left] {{\begin{color}{green} $\tau_1:$\end{color}}};
		  
		
        
          
          \draw ([c]O2)-- ([c]B1);
          \draw ([c]O2)-- ([c]B2);
          \draw ([c]B1)-- ([c]C1);
          \draw ([c]B2)-- ([c]B2l);
        
          
         
          \node at (14.5,7.7,0) [left] {{\begin{color}{blue}\scriptsize{$2$}\end{color}}};
           \node at (15.3,7.7,0) [right] {{\begin{color}{blue}\scriptsize{$1$}\end{color}}};
          
          
           
          
           \draw[ thick] ([c]y1)-- ([c]y2);
            \draw[thick] ([c]x1)-- ([c]x2);
            
              \draw [gray]([c]oo) parabola ([c]m1);
               \draw[gray] ([c]oo) parabola ([c]m2); 
            
           
            \node at ([c]m1) [right] {{\scriptsize{$\overline{T}$}}};
             \node at ([c]x2) [right] {{\scriptsize{$T$}}};
              \node at ([c]y2) [right] {{\scriptsize{$W$}}};
              
            
                
		  \draw[fill=white]  ([c]O2) circle (3pt);
		  
		
          \fill ([c]B2) circle (3pt);
           \draw[fill=red] ([c]B1) circle (3pt);
           
            \end{scope}

		
		
		 \begin{scope}[every coordinate/.style={shift={(15,4,0)}}]
		  
		  \node at (12,4,0) [left] {{$\sigma_2:$}};
		  
		
          
          \draw ([c]O22)-- ([c]B2l);
          \draw ([c]O22)-- ([c]B12);
          \draw ([c]B12)-- ([c]C12);
           \draw ([c]C12)-- ([c]C1);
          \draw ([c]B2l)-- ([c]C2);
        
         
          
          
            \node at (14.4,4,0) [left] {{\begin{color}{blue}\scriptsize{$2$}\end{color}}};
              \node at (13.9,3.3,0) [left] {{\begin{color}{blue}\scriptsize{$1$}\end{color}}};
              \node at (15.5,4,0) [right] {{\begin{color}{blue}\scriptsize{$1$}\end{color}}};
          
          
         
          \draw [decorate,decoration={brace,amplitude=5pt,mirror},xshift=-0.4pt] ([c]B12)--([c]O22) node[black,midway,xshift=0.4cm,yshift=-0.85cm] {\scriptsize{$l_2-l_1$}};
           \draw [decorate,decoration={brace,amplitude=5pt,mirror},xshift=-0.4pt,yshift=-0.4pt] ([c]C12)--([c]B12) node[black,midway,xshift=-0.2cm,yshift=-0.4cm] {\scriptsize{$2l_1-l_2$}};
           
          
           
          
           \draw[gray] ([c]y1)-- ([c]y1e)--([c]y2e)--([c]y2)--([c]y1);
            \draw[thick] ([c]b1)-- ([c]b2);
            \draw[thick] ([c]b1e)-- ([c]b2e);
              
            
           
        
             \node at ([c]b2) [right] {{\scriptsize{$T$}}};
              \node at ([c]b2e) [right] {{\scriptsize{$W$}}};
              
             
                
		  \draw[fill=white]  ([c]O22) circle (3pt);
		  
			\foreach \x in {C12,B2l} 
          \fill ([c]\x) circle (3pt);
          
         \draw ([c]B12) node[red,cross]{};
              
              
            \end{scope}

		\begin{scope}[every coordinate/.style={shift={(15,0,0)}}]
		  
		 \node at (12,0,0) [left] {{\begin{color}{orange} $\tau_2:$\end{color}}};
		  
		  
		
      
           \draw ([c]O)-- ([c]B2);
          \draw ([c]O)-- ([c]Bt1);
          \draw ([c]Bt1)-- ([c]C1);
       \draw ([c]B2)-- ([c]C2);
        
         
          
          
            \node at (14.5,-.7,0) [left] {{\begin{color}{blue}\scriptsize{$1$}\end{color}}};
            \node at (15.5,-.7,0) [right] {{\begin{color}{blue}\scriptsize{$1$}\end{color}}};
          
          
          
           
          
           
          
           \draw[gray] ([c]y1)-- ([c]y1e)--([c]y2e)--([c]y2)--([c]y1);
            \draw[thick] ([c]b1)-- ([c]b2);
            \draw[thick] ([c]b1e)-- ([c]b2e);
              
            
           
        
             \node at ([c]b2) [right] {{\scriptsize{$T$}}};
              \node at ([c]b2e) [right] {{\scriptsize{$W$}}};
              
           
              \draw[fill=red]  ([c]O) circle (3pt);
		  
			\foreach \x in {Bt1,B2} 
          \fill ([c]\x) circle (3pt);
          
              
            \end{scope}
		  \begin{scope}[every coordinate/.style={shift={(15,-4,0)}}]
		  
		  \node at (12,-4,0) [left] {{$\sigma_3:$}};
		  
		 
          
          \draw ([c]O3)-- ([c]Bt1);
          \draw ([c]O3)-- ([c]D2);
          \draw ([c]Bt1)-- ([c]C12);
          \draw ([c]D2)-- ([c]B2l);
          \draw ([c]B2l)-- ([c]C2);
          
         
          \node at (14.5,-4.5,0) [left] {{\begin{color}{blue}\scriptsize{$1$}\end{color}}};
           \node at (15.6,-3.5,0) {{\begin{color}{blue}\scriptsize{$3$}\end{color}}};
            \node at (15.9,-5.05,0) [above right] {{\begin{color}{blue}\scriptsize{$1$}\end{color}}};
          
         
          \draw [decorate,decoration={brace,amplitude=5pt},xshift=0.9pt,yshift=-0.4pt] ([c]D2)--([c]O3) node[black,midway,xshift=-0.1cm,yshift=-0.95cm] {\scriptsize{$\frac{l_1-l_2}{2}$}};
           \draw [decorate,decoration={brace,amplitude=5pt,mirror},xshift=0.2pt,yshift=-0.3pt] ([c]D2)--([c]B2l) node[black,midway,xshift=0.15cm,yshift=-0.47cm] {\scriptsize{$\frac{3l_2
           -l_1}{2}$}};
           
             
          
           \draw[gray] ([c]y1)-- ([c]y1e)--([c]y2e)--([c]y2)--([c]y1);
            \draw[thick] ([c]b1)-- ([c]b2);
            \draw[thick] ([c]b1e)-- ([c]b2e);
              
            
           
        
             \node at ([c]b2) [right] {{\scriptsize{$T$}}};
              \node at ([c]b2e) [right] {{\scriptsize{$W$}}};
              
           
             
		  \draw[fill=white]  ([c]O3) circle (3pt);
		  
		
          \fill ([c]Bt1) circle (3pt);
           \fill ([c]B2l) circle (3pt);
           
           
        
             \draw ([c]D2) node[cross,red]{};
              
              
            \end{scope}
	

		  \begin{scope}[every coordinate/.style={shift={(15,-8,0)}}]
		  
		  \node at (12,-8,0) [left] {{\begin{color}{red} $\tau_3:$\end{color}}};
		  
		
          
          \draw ([c]O3)-- ([c]Bt1);
          \draw ([c]O3)-- ([c]D22);
          \draw ([c]Bt1)-- ([c]C1);
          \draw ([c]D22)-- ([c]B2l);
        
          
         
          \node at (14.5,-8.3,0) [left] {{\begin{color}{blue}\scriptsize{$1$}\end{color}}};
           \node at (15.3,-8.3,0) [right] {{\begin{color}{blue}\scriptsize{$3$}\end{color}}};
          
          
           
          
           \draw[ thick] ([c]y1)-- ([c]y2);
           
            
              \draw[thick] ([c]oo) parabola ([c]m1);
               \draw[thick] ([c]oo) parabola ([c]m3); 
            
           
            \node at ([c]m1) [right] {{\scriptsize{$W$}}};
          
              \node at ([c]y2) [right] {{\scriptsize{$T$}}};
              
            
		  \draw[fill=white]  ([c]O3) circle (3pt);
		  
		
          \fill ([c]Bt1) circle (3pt);
           \draw[fill=red] ([c]D22) circle (3pt);
          
              
            \end{scope}
            
           
		  \begin{scope}[every coordinate/.style={shift={(15,-12,0)}}]
		  
		  \node at (12,-12,0) [left] {{$\sigma_4:$}};
		  
		  
		 
		  
		
      
          \draw ([c]O3)-- ([c]Bt1);
          \draw ([c]O3)-- ([c]D22);
          \draw ([c]Bt1)-- ([c]C12);
           \draw ([c]C12)-- ([c]C1);
          \draw ([c]D22)-- ([c]B2l);
          
          
         
          \node at (14.5,-12.2,0) [left] {{\begin{color}{blue}\scriptsize{$1$}\end{color}}};
           
           \node at (15.3,-12.2,0) [right] {{\begin{color}{blue}\scriptsize{$3$}\end{color}}};
           
          
         
          \draw [decorate,decoration={brace,amplitude=5pt,mirror},xshift=0.2pt,yshift=-0.4pt] ([c]Bt1)--([c]O3) node[black,midway,xshift=0.2cm,yshift=-.88cm] {\scriptsize{$3  l_2$}};
           \draw [decorate,decoration={brace,amplitude=5pt},xshift=0.2pt,yshift=-0.3pt] ([c]Bt1)--([c]C12) node[black,midway,xshift=-.2cm,yshift=-0.4cm] {\scriptsize{$l_1 - 3l_2$}};
           
             
           
           \draw[gray] ([c]y1)-- ([c]y2);
            \draw[thick] ([c]b1)-- ([c]b2);
            
              \draw[thick] ([c]oo) parabola ([c]m1);
               \draw[thick] ([c]oo) parabola ([c]m3); 
            
           
            \node at ([c]m1) [right] {{\scriptsize{$W$}}};
          
              \node at ([c]b2) [right] {{\scriptsize{$T$}}};
          
         
           \fill ([c]C12) circle (3pt);
           \draw[fill=red] ([c]D22) circle (3pt);
           
          \draw[fill=white]  ([c]O3) circle (3pt);
         \draw ([c]Bt1) node[cross]{};
        
          
          
          \end{scope}
	

		
		
	\end{tikzpicture}
	\caption{Tropical admissible cover (top left); the subdivision of $\operatorname{trop}(\mathcal A_2^{\text{wt}})$ (bottom left);  $\lambda_T$  and the  Gorenstein singularities (right).}
 \label{esesub1}
\end{figure}
\end{exa}


\begin{exa}
The stabilisation of $C$ has a smooth core of weight zero, and three rational tails attached to general points. Aligning with respect to all the admissible functions produces the non-simplicial subdivision (a); the subdivisions (b) and (c) are the coarsening in case of high-weight, respectively weight-one, tails.
We also represent $\lambda_T$ and the associated singularity on some cones of the subdivision; the other ones can be derived by symmetry. In case (c) the singularity over $\sigma_2$ is replaced by a (sprouted) ribbon. See Figure \ref{esesub2}.

\begin{figure}[h]
\begin{tikzpicture} [scale=.4]
\tikzstyle{every node}=[font=\normalsize]
		\tikzset{arrow/.style={latex-latex}}
		
	\tikzset{cross/.style={cross out, draw, thick,
         minimum size=2*(#1-\pgflinewidth), 
         inner sep=1.2pt, outer sep=1.2pt}}
         
         
        
         
         
        
		\coordinate (Oc) at (0,3,0);
		\coordinate (T1) at (-2,4,0);
		\coordinate (bT1) at (-2,2,0);
		\coordinate (T2) at (4,1.5,0);
		\coordinate (bT2) at (4,2.5,0);
		\coordinate (T3) at (5,6.5,5);
		\coordinate (bT3) at (5,7.5,5);
	
		\coordinate (Ot) at (0,0,0);
		\coordinate (T1t) at (-2,0,0);
		\coordinate (T2t) at (4,0,0);
		\coordinate (T3t) at (5,0,5);
	
		
		\coordinate (P1) at (-.5,-1,0);
		\coordinate (P2) at (0,-1,0);
		\coordinate (P3) at (.5,-1,0);
		\coordinate (P4) at (1,-1,0);
		\coordinate (P5) at (1.8,-.7,0);
    	\coordinate (P6) at (-1,-.8,0);
		
	
		
		\coordinate (l1) at (-1.2,2.3,0);
		\coordinate (bl1) at (-.9,4.4,0);
		\coordinate (l2) at (2.5,1.8,0);
		\coordinate (bl2) at (2.5,2.5,0);
		\coordinate (l3) at (3.6,4.4,3);
		\coordinate (bl3) at (3,5.8,3);
	
		\coordinate (Ot) at (0,0,0);
		\coordinate (l1t) at (-1,0.2,0);
		\coordinate (l2t) at (1.5,0,0);
		\coordinate (l3t) at (3.5,0,3);
			
		\begin{scope}[every coordinate/.style={shift={(-6,10,0)}}]
		
		
     
     	
     	\draw[dotted] ([c]Oc)-- ([c]T1);
     	\draw ([c]Oc)-- ([c]bT1);
     	\draw ([c]Oc)-- ([c]T2);
     	\draw[dotted] ([c]Oc)-- ([c]bT2);
     	\draw ([c]Oc)-- ([c]T3);
     	\draw[dotted] ([c]Oc)-- ([c]bT3);
     	
     	\draw ([c]Ot)-- ([c]T3t);
     	\draw ([c]Ot)-- ([c]T2t);
     	\draw ([c]Ot)-- ([c]T1t);
     	
     
     	\draw[blue] ([c]Ot)--([c]P1);
     	\draw[blue] ([c]Ot)--([c]P2);
     	\draw[blue] ([c]Ot)--([c]P3);
     	\draw[blue] ([c]Ot)--([c]P4);
     	\draw[blue] ([c]Ot)--([c]P5);
     	\draw[blue] ([c]Ot)--([c]P6);

      
       
       \node at ([c]l1) [above] {\scriptsize{$l_1$}};
      
        \node at ([c]l2) [above] {\scriptsize{$l_2$}};
       
           \node at ([c]l3) [above=.4cm] {\scriptsize{$l_3$}};
       
        
      \node at ([c]l1t) [above] {\scriptsize{$l_1$}};
       \node at ([c]l2t) [above] {\scriptsize{$l_2$}};
     
           \node at ([c]l3t) [above] {\scriptsize{$l_3$}};
        
       
      \node at ([c]T1) [left] {\scriptsize{$\overline{T}_1$}};
        \node at ([c]bT1) [left] {\scriptsize{$T_1$}};
         
        \node at ([c]bT2) [right] {\scriptsize{$\overline{T}_2$}};
        \node at ([c]T2) [right] {\scriptsize{$T_2$}};
         
         \node at ([c]bT3) [right] {\scriptsize{$\overline{T}_3$}};
        \node at ([c]T3) [right] {\scriptsize{$T_3$}};
       
      
       \draw[dashed] ([c]Oc)--([c]Ot);
       \draw[dashed] ([c]T1)--([c]T1t);
       \draw[dashed] ([c]bT2)--([c]T2t);
       \draw[dashed] ([c]bT3)--([c]T3t);
      
     
		\draw[fill=white]  ([c]Oc) circle (3pt);
			\draw  ([c]T1) node [cross]{};
			\draw  ([c]bT2) node [cross]{};
    \draw  ([c]bT3) node [cross]{};

    			\draw  ([c]Ot) circle (3pt);

						
						\foreach \x in {bT1,T2,T3,T1t,T2t,T3t} 
          \fill ([c]\x) circle (3pt);
      
         \end{scope}
        
         \coordinate(label) at (0,-.5,0);
         \coordinate (A) at (-5,0,0);
         \coordinate (A1) at (-2.5,0,0);
         \coordinate (A2) at (0,0,0);
        \coordinate (A3) at (2.5,0,0);
        \coordinate (B) at (5,0,0);
         \coordinate (B1) at (2.5,4.3,0);
         \coordinate (B2) at (3.75,2.15,0);
         \coordinate (B3) at (1.25,6.45,0);
         \coordinate (C) at (0,8.6,0);
         \coordinate (C1) at (-2.5,4.3,0);
         \coordinate (C2) at (-3.75,2.15,0);
         \coordinate (C3) at (-1.25,6.45,0);
         \coordinate (M1) at (0,2.87,0);
           \coordinate (M2) at (0,1.23,0);
          \coordinate (M3) at (1.43,3.69,0);
        \coordinate (M3b) at (-1.43,3.69,0);
        
         \coordinate (MM) at (0,2,0);
          \coordinate (MA1) at (-2.6,1.5,0);
         \coordinate (MA2) at (-1.8,.8,0);
         \coordinate (CoB) at (-2,1.6,0);
         \coordinate (CoA) at (-.3,.8,0);
         
        
       \begin{scope} [every coordinate/.style={shift={(-20,-2,0)}}]
       
        \node at ([c]label)[below]  {(a)};
        
        \draw ([c]A)--([c]B);
         \draw ([c]A)--([c]C);
          \draw ([c]C)--([c]B);
          
         \draw([c]C)--([c]A1); 
         \draw([c]C)--([c]A2); 
        \draw([c]C)--([c]A3); 
        
        \draw([c]A)--([c]B1); 
         \draw([c]A)--([c]B2); 
        \draw([c]A)--([c]B3); 
        
        \draw ([c]B)--([c]C1); 
         \draw([c]B)--([c]C2); 
        \draw([c]B)--([c]C3); 
        
        \node at ([c]C) [above] {\scriptsize{$l_3$}};
        \node at ([c]A) [left] {\scriptsize{$l_1$}};
        \node at ([c]B) [right] {\scriptsize{$l_2$}};
        
          \fill ([c]M1) circle (2pt);
           \fill ([c]M2) circle (2pt);
             \fill ([c]M3) circle (2pt);
              \fill ([c]M3b) circle (2pt);
        
        \end{scope}
        
       
        \begin{scope} [every coordinate/.style={shift={(-6,-2,0)}}]
        
        \node at ([c]label)[below]  {(b)};
        
        \draw ([c]A)--([c]B);
         \draw ([c]A)--([c]C);
          \draw ([c]C)--([c]B);
          
          \draw[red]([c]M1)--([c]A);
           \draw[]([c]M1)--([c]B);
            \draw[orange]([c]M2)--([c]A);
           \draw[]([c]M2)--([c]B);
          \draw[green] ([c]M2)--([c]M1);
           
         \draw([c]C)--([c]M2); 
         \draw([c]M2)--([c]A); 
        \draw([c]M2)--([c]B); 
        
        \draw([c]A)--([c]M3); 
         \draw([c]M3)--([c]B); 
        \draw([c]M3)--([c]C); 
        
        \draw ([c]B)--([c]M3b); 
         \draw([c]M3b)--([c]A); 
        \draw([c]M3b)--([c]C); 
        
        \node at ([c]C) [above] {\scriptsize{$l_3$}};
        \node at ([c]A) [left] {\scriptsize{$l_1$}};
        \node at ([c]B) [right] {\scriptsize{$l_2$}};
        
        
              
             \draw[thick] ([c]A)--([c]B);
           \fill[red] ([c]M1) circle (2pt);
           \fill[orange] ([c]M2) circle (2pt);
           \draw[thick]([c]M1)--([c]A);
           \draw[thick]([c]M1)--([c]B);
            \draw[blue]([c]M2)--([c]A);
           \draw[]([c]M2)--([c]B);
          \draw[green] ([c]M2)--([c]M1);
         
          \node at ([c]M1) [above right] {\begin{color}{red}\scriptsize{$r_1$}\end{color}};
         \node at ([c]M2) [above right] {\begin{color}{orange}\scriptsize{$r_2$}\end{color}};
          \node at ([c]MM) [left=-.1cm] {\begin{color}{green}\scriptsize{$\tau_2$}\end{color}};
           
                \node at ([c]MA2) [below] {\begin{color}{blue}\scriptsize{$\tau_3$}\end{color}};
          \node at ([c]CoB) [right] {\scriptsize{$\sigma_1$}};
           \node at ([c]CoA) [below] {\scriptsize{$\sigma_2$}};
      
              \fill[red] ([c]M1) circle (3pt);
           \fill[orange] ([c]M2) circle (3pt);
             \fill ([c]M3) circle (2pt);
              \fill ([c]M3b) circle (2pt);

        \end{scope}
        
       
        
        \begin{scope} [every coordinate/.style={shift={(8,-2,0)}}]
        
         \node at ([c]label)[below]  {(c)};
        
        \draw ([c]A)--([c]B);
         \draw ([c]A)--([c]C);
          \draw ([c]C)--([c]B);
          
         \draw([c]C)--([c]M2); 
         \draw([c]M2)--([c]A); 
        \draw([c]M2)--([c]B); 
            \draw([c]M2)--([c]A2); 
        
        \draw([c]A)--([c]M3); 
         \draw([c]M3)--([c]B); 
        \draw([c]M3)--([c]C); 
           \draw([c]M3)--([c]B1); 
        
        \draw ([c]B)--([c]M3b); 
         \draw([c]M3b)--([c]A); 
        \draw([c]M3b)--([c]C); 
         \draw([c]M3b)--([c]C1); 
        
        \node at ([c]C) [above] {\scriptsize{$l_3$}};
        \node at ([c]A) [left] {\scriptsize{$l_1$}};
        \node at ([c]B) [right] {\scriptsize{$l_2$}};
        
          \fill ([c]M1) circle (2pt);
           \fill ([c]M2) circle (2pt);
             \fill ([c]M3) circle (2pt);
              \fill ([c]M3b) circle (2pt);
              
              
              
        
        \end{scope}
         
         
        
         \coordinate(cone1) at (2.8,2.2,0);
        \coordinate(cone) at (3.2,2.5,0);
         \coordinate (o) at (0,0,0);
          \coordinate (t3) at (-1,-3.3,0);
           \coordinate (t2) at (1,-3.3,0);
          \coordinate (t1) at (2,-3,-1);
          \coordinate (t3p) at (-2.5,-3.3,0);
           \coordinate (t2p) at (2.5,-3.3,0);
          \coordinate (t1p) at (3.5,-3,-1);
          \coordinate (t1plong) at (4.5,-3,-1);
         
          \coordinate (t3halfrip) at (-.5,-2.5,-1);
       
          \coordinate (t3rip) at (-.2,-3.3,0);
         
          \coordinate (s3) at (-.5,-1.8,0);
           \coordinate (s3rip) at (-.2,-1.8,0);
          \coordinate (s2) at (.5,-1.8,0);
          \coordinate (s1) at (1,-1.4,-.5);
           
            \coordinate (s31) at (-.2,-2.5,0);
           \coordinate (s3halfrip) at (-.3,-.7,0);
            
           
            \coordinate (t2plong) at (3.5,-3.3,0);
            
           
      \coordinate (y1) at (5.5,0,0);
	   \coordinate (y2) at (5.5,-3.4,0);
	     \coordinate (b1) at (5.2,-.4,0);
	    \coordinate (b2) at (6.8,.2,0);
	     \coordinate (b1e) at (6.2,.3,0);
	    \coordinate (b2e) at (7.8,-.5,0);
	     \coordinate (y1e) at (5.7,0,0);
	   \coordinate (y2e) at (5.7,-3.4,0);
	   \coordinate (c1) at (5.2,-3,0);
	    \coordinate (c2) at (6.8,-2.4,0);
	    \coordinate (d1) at (5.2,-1.6,0);
	    \coordinate (d2) at (6.8,-1,0);
            
               
            
          \coordinate (x1) at (5,-1.5,0);
	   \coordinate (x2) at (8,-1.5,0);
	   \coordinate (oo) at(6.2,-1.5,0);
	    \coordinate (m1) at (8.4,-1.8,2);
	   \coordinate (m2) at (5.7,-1.8,2);
	   \coordinate (z1) at (5.4,-.2,0);
	   \coordinate (z2) at (7,-3,0);
	   \coordinate (w1) at (7,0.2,0);
	   \coordinate (w2) at (5.4,-3,0);
	      
	       
	        \coordinate (u1) at (6.4,-.1,0);
	   \coordinate (u2) at (8.4,-.5,0);
	   
	    \coordinate (v1) at (6.3,-2.5,0);
	   \coordinate (v2) at (8.5,-3,0);
          
          \begin{scope}[every coordinate/.style={shift={(-18,-7,0)}}]
          
           \node at ([c]cone) [left] {\begin{color}{red}$r_1:$\end{color}}; 
             \node at ([c]cone1) [below] {\scriptsize{$l_1=l_2=l_3$}}; 
           
            
            \draw ([c]o)--([c]t1) node[above]{\scriptsize{$T_2$}};
            \draw ([c]o)--([c]t2) node[below]{\scriptsize{$T_1$}};
            \draw ([c]o)--([c]t3) node[below]{\scriptsize{$T_3$}};
              
              \draw ([c]t1)--([c]t1p);
              \draw ([c]t2)--([c]t2p);
              \draw ([c]t3)--([c]t3p);
             
         \node at ([c]s1) [above] {\begin{color}{blue}\scriptsize{$1$}\end{color}};   
          \node at ([c]s2) [right] {\begin{color}{blue}\scriptsize{$1$}\end{color}}; 
           \node at ([c]s3) [left] {\begin{color}{blue}\scriptsize{$1$}\end{color}}; 
           
           
           
           \draw[gray] ([c]y1)-- ([c]y1e)--([c]y2e)--([c]y2)--([c]y1);
            \draw[thick] ([c]b1)-- ([c]b2);
            \draw[thick] ([c]c1)-- ([c]c2);
              \draw[thick] ([c]d1)-- ([c]d2);
              
           
            \node at ([c]b2) [right] {\scriptsize{$T_1$}};
            \node at ([c]c2) [right] {\scriptsize{$T_3$}};   
            \node at ([c]d2) [right] {\scriptsize{$T_2$}};   
            
           
            \draw[fill=red] ([c]o) circle (3pt);
           \fill ([c]t1) circle (3pt);
             \fill ([c]t2) circle (3pt);
              \fill ([c]t3) circle (3pt);
              
          \end{scope}
          
           \begin{scope}[every coordinate/.style={shift={(-2,-7,0)}}]
          
           \node at ([c]cone) [left] {\begin{color}{orange}$r_2:$\end{color}}; 
             \node at ([c]cone1) [below] {\scriptsize{$l_1=l_2=2l_3$}}; 
           
            
            \draw ([c]o)--([c]t1);
            \draw ([c]o)--([c]t2);
            \draw ([c]o)--([c]t3rip);
              
              \draw ([c]t1)--([c]t1p);
              \draw ([c]t2)--([c]t2p);
              \draw ([c]t3rip)--([c]t3p);
             
         \node at ([c]s1) [above] {\begin{color}{blue}\scriptsize{$1$}\end{color}};   
          \node at ([c]s2) [right] {\begin{color}{blue}\scriptsize{$1$}\end{color}}; 
           \node at ([c]s3rip) [left] {\begin{color}{blue}\scriptsize{$2$}\end{color}}; 
           
           
                  
             \draw[fill=white] ([c]o) circle (3pt);
           \fill ([c]t1) circle (3pt);
             \fill ([c]t2) circle (3pt);
              \draw[fill=red] ([c]t3rip) circle (3pt);
              
           \end{scope}
           
            \begin{scope}[every coordinate/.style={shift={(0,-7,0)}}]
           
           
           
            \draw[thick] ([c]x1)--([c]x2);
                \draw[thick] ([c]z1)--([c]z2);
                \draw[thick] ([c]w1)--([c]w2);
              \draw[gray] ([c]oo) parabola ([c]m1);
               \draw[gray] ([c]oo) parabola ([c]m2); 
              
           
            \node at ([c]w1) [right] {\scriptsize{$T_1$}};
            \node at ([c]x2) [right] {\scriptsize{$T_3$}};   
            \node at ([c]z1) [left] {\scriptsize{$T_2$}};   
              
     
              
          \end{scope}
         
          \begin{scope}[every coordinate/.style={shift={(-18,-15,0)}}]
          
           \node at ([c]cone) [left] {\begin{color}{green}$\tau_2:$\end{color}}; 
             \node at ([c]cone1) [below] {\scriptsize{$l_3\leq l_1=l_2\leq 2l_3$}}; 
          
            
            \draw ([c]o)--([c]t1);
            \draw ([c]o)--([c]t2);
            \draw ([c]o)--([c]t3halfrip);
            \draw ([c]t3)--([c]t3halfrip);
              
              \draw ([c]t1)--([c]t1p);
              \draw ([c]t2)--([c]t2p);
              \draw ([c]t3)--([c]t3p);
             
         \node at ([c]s1) [above] {\begin{color}{blue}\scriptsize{$1$}\end{color}};   
          \node at ([c]s2) [right] {\begin{color}{blue}\scriptsize{$1$}\end{color}}; 
           \node at ([c]s3halfrip) [left] {\begin{color}{blue}\scriptsize{$2$}\end{color}}; 
            \node at ([c]s31) [below] {\begin{color}{blue}\scriptsize{$1$}\end{color}}; 
            
            
             \draw [decorate,decoration={brace,amplitude=5pt,mirror},xshift=0.2pt,yshift=-0.4pt] ([c]o)--([c]t3halfrip) node[black,midway,xshift=-0.6cm,yshift=-0.2cm] {\scriptsize{$l_2-l_3$}};
           \draw [decorate,decoration={brace,amplitude=5pt,mirror},xshift=0.2pt,yshift=-0.3pt] ([c]t3halfrip)--([c]t3) node[black,midway,xshift=-0.7cm,yshift=0.1cm] {\scriptsize{$2l_3-l_2$}};
         
            
            
            
            
            
           
           \draw[gray] ([c]y1)-- ([c]y1e)--([c]y2e)--([c]y2)--([c]y1);
            \draw[thick] ([c]b1)-- ([c]b2);
            \draw[thick] ([c]c1)-- ([c]c2);
              \draw[thick] ([c]d1)-- ([c]d2);
              
           
            \node at ([c]b2) [right] {\scriptsize{$T_1$}};
            \node at ([c]c2) [right] {\scriptsize{$T_3$}};   
            \node at ([c]d2) [right] {\scriptsize{$T_2$}};   
           
           
             \draw[fill=white] ([c]o) circle (3pt);
           \fill ([c]t1) circle (3pt);
             \fill ([c]t2) circle (3pt);
              \fill ([c]t3) circle (3pt);
            
             \draw ([c]t3halfrip) node[red,cross]{};
            
            
          \end{scope}
         
       
       
        \begin{scope}[every coordinate/.style={shift={(-2,-15,0)}}]
          
           \node at ([c]cone) [left] {\begin{color}{blue}$\tau_3:$\end{color}}; 
             \node at ([c]cone1) [below] {\scriptsize{$l_1\geq l_2=2l_3$}}; 
           
            
            
            \draw ([c]o)--([c]t1);
            \draw ([c]o)--([c]t2);
            \draw ([c]o)--([c]t3rip);
              
              \draw ([c]t1)--([c]t1p);
              \draw ([c]t2)--([c]t2p);
               \draw ([c]t2p)--([c]t2plong);
              \draw ([c]t3rip)--([c]t3p);
             
         \node at ([c]s1) [above] {\begin{color}{blue}\scriptsize{$1$}\end{color}};   
          \node at ([c]s2) [right] {\begin{color}{blue}\scriptsize{$1$}\end{color}}; 
           \node at ([c]s3rip) [left] {\begin{color}{blue}\scriptsize{$2$}\end{color}}; 
           
           
            \draw [decorate,decoration={brace,amplitude=5pt,mirror},xshift=0.2pt,yshift=-0.4pt] ([c]o)--([c]t2) node[black,midway,xshift=-0.1cm,yshift=-0.4cm] {\scriptsize{$l_2$}};
           \draw [decorate,decoration={brace,amplitude=5pt,mirror},xshift=0.2pt,yshift=-0.3pt] ([c]t2)--([c]t2p) node[black,midway,xshift=0cm,yshift=-0.3cm] {\scriptsize{$l_1-l_2$}};
         
         
           
             \draw[fill=white] ([c]o) circle (3pt);
           \fill ([c]t1) circle (3pt);
             \fill ([c]t2p) circle (3pt);
              \fill[red] ([c]t3rip) circle (3pt);
              
               \draw ([c]t2) node[cross]{};
           \end{scope}
           \begin{scope}[every coordinate/.style={shift={(0,-15,0)}}]
           
           
           
            \draw[thick] ([c]x1)--([c]x2);
                \draw[thick] ([c]z1)--([c]z2);
                 \draw[thick] ([c]u1)--([c]u2);
                \draw[gray] ([c]w1)--([c]w2);
              \draw[gray] ([c]oo) parabola ([c]m1);
               \draw[gray] ([c]oo) parabola ([c]m2); 
              
           
            \node at ([c]w1) [right] {\scriptsize{$T_1$}};
            \node at ([c]x2) [right] {\scriptsize{$T_3$}};   
            \node at ([c]z1) [left] {\scriptsize{$T_2$}};   
              
          
              
          \end{scope}
         
        \begin{scope}[every coordinate/.style={shift={(-18,-23,0)}}]
  
          
           \node at ([c]cone) [left] {$\sigma_1$:}; 
             \node at ([c]cone1) [below] {\scriptsize{$l_3\leq l_2\leq l_1, 2l_3$}}; 
           
               
            \draw ([c]o)--([c]t1);
            \draw ([c]o)--([c]t2);
            \draw ([c]o)--([c]t3halfrip);
            \draw ([c]t3)--([c]t3halfrip);
              
              \draw ([c]t1)--([c]t1p);
              \draw ([c]t2)--([c]t2p);
               \draw ([c]t2p)--([c]t2plong);
              \draw ([c]t3)--([c]t3p);
             
         \node at ([c]s1) [above] {\begin{color}{blue}\scriptsize{$1$}\end{color}};   
          \node at ([c]s2) [right] {\begin{color}{blue}\scriptsize{$1$}\end{color}}; 
           \node at ([c]s3halfrip) [left] {\begin{color}{blue}\scriptsize{$2$}\end{color}}; 
            \node at ([c]s31) [below] {\begin{color}{blue}\scriptsize{$1$}\end{color}}; 
            
            
             \draw [decorate,decoration={brace,amplitude=5pt,mirror},xshift=0.2pt,yshift=-0.4pt] ([c]o)--([c]t3halfrip) node[black,midway,xshift=-0.6cm,yshift=-0.2cm] {\scriptsize{$l_2-l_3$}};
           \draw [decorate,decoration={brace,amplitude=5pt,mirror},xshift=0.2pt,yshift=-0.3pt] ([c]t3halfrip)--([c]t3) node[black,midway,xshift=-0.7cm,yshift=0.1cm] {\scriptsize{$2l_3-l_2$}};
         
            
            \draw [decorate,decoration={brace,amplitude=5pt,mirror},xshift=0.2pt,yshift=-0.4pt] ([c]o)--([c]t2) node[black,midway,xshift=-0.1cm,yshift=-0.4cm] {\scriptsize{$l_2$}};
           \draw [decorate,decoration={brace,amplitude=5pt,mirror},xshift=0.2pt,yshift=-0.3pt] ([c]t2)--([c]t2p) node[black,midway,xshift=0cm,yshift=-0.3cm] {\scriptsize{$l_1-l_2$}};
            
               
           \draw[fill=white] ([c]o) circle (3pt);
           \fill ([c]t1) circle (3pt);
             \fill ([c]t2p) circle (3pt);
              \fill ([c]t3) circle (3pt);
            
             \draw ([c]t3halfrip) node[red,cross]{};
               \draw ([c]t2) node[cross]{};
            
            
            
           
           \draw[gray] ([c]y1)-- ([c]y1e)--([c]y2e)--([c]y2)--([c]y1);
            \draw[thick] ([c]b1e)-- ([c]b2e);
                 \draw[gray] ([c]b1)-- ([c]b2);
            \draw[thick] ([c]c1)-- ([c]c2);
              \draw[thick] ([c]d1)-- ([c]d2);
              
           
            \node at ([c]b2e) [right] {\scriptsize{$T_1$}};
            \node at ([c]c2) [right] {\scriptsize{$T_3$}};   
            \node at ([c]d2) [right] {\scriptsize{$T_2$}};   
           
       
          \end{scope}
         
         
        \begin{scope}[every coordinate/.style={shift={(-2,-23,0)}}]
          
           \node at ([c]cone) [left] {$\sigma_2$:}; 
             \node at ([c]cone1) [below] {\scriptsize{$l_1,l_2\geq 2l_3$}}; 
           
            
            \draw ([c]o)--([c]t1);
            \draw ([c]o)--([c]t2);
            \draw ([c]o)--([c]t3rip);
              
              \draw ([c]t1)--([c]t1p);
              \draw ([c]t2)--([c]t2p);
               \draw ([c]t2p)--([c]t2plong);
                \draw ([c]t1p)--([c]t1plong);
              \draw ([c]t3rip)--([c]t3p);
             
         \node at ([c]s1) [above] {\begin{color}{blue}\scriptsize{$1$}\end{color}};   
          \node at ([c]s2) [right] {\begin{color}{blue}\scriptsize{$1$}\end{color}}; 
           \node at ([c]s3rip) [left] {\begin{color}{blue}\scriptsize{$2$}\end{color}}; 
           
           
            \draw [decorate,decoration={brace,amplitude=5pt,mirror},xshift=0.2pt,yshift=-0.4pt] ([c]o)--([c]t2) node[black,midway,xshift=-0.1cm,yshift=-0.4cm] {\scriptsize{$2l_3$}};
           \draw [decorate,decoration={brace,amplitude=5pt,mirror},xshift=0.2pt,yshift=-0.3pt] ([c]t2)--([c]t2p) node[black,midway,xshift=0cm,yshift=-0.3cm] {\scriptsize{$l_1-2l_3$}};
           
           \draw [decorate,decoration={brace,amplitude=5pt,mirror},xshift=0.2pt,yshift=-0.4pt] ([c]t1)--([c]o) node[black,midway,xshift=0.4cm,yshift=0.2cm] {\scriptsize{$2l_3$}};
           \draw [decorate,decoration={brace,amplitude=5pt,mirror},xshift=0.2pt,yshift=-0.3pt] ([c]t1p)--([c]t1) node[black,midway,xshift=0.2cm,yshift=0.4cm] {\scriptsize{$l_2-2l_3$}};
         
          
           \draw[fill=white] ([c]o) circle (3pt);
           
             \fill ([c]t2p) circle (3pt);
              \fill ([c]t1p) circle (3pt);
              \fill[red] ([c]t3rip) circle (3pt);
              
               \draw ([c]t2) node[cross]{};
            \draw ([c]t1) node[cross]{};
            
            \end{scope}
             \begin{scope}[every coordinate/.style={shift={(0,-23,0)}}]
           
           
            \draw[thick] ([c]x1)--([c]x2);
                \draw[gray] ([c]z1)--([c]z2);
                 \draw[thick] ([c]u1)--([c]u2);
                   \draw[thick] ([c]v1)--([c]v2);
                \draw[gray] ([c]w1)--([c]w2);
              \draw[gray] ([c]oo) parabola ([c]m1);
               \draw[gray] ([c]oo) parabola ([c]m2); 
              
           
            \node at ([c]w1) [right] {\scriptsize{$T_1$}};
            \node at ([c]x2) [right] {\scriptsize{$T_3$}};   
            \node at ([c]v2) [right] {\scriptsize{$T_2$}};   
              
          \end{scope}
       
       \end{tikzpicture}
	\caption{(a) Subdivision $\Sigma$ (non-simplicial). (b) Subdivision $\Sigma'$  for high-degree tails. (c) Subdivision $\Sigma'$  for degree-one tails.}
 \label{esesub2}
\end{figure}
\end{exa}

\begin{exa}
The core of $C$ consists of two elliptic curves of weight zero, meeting in a node; each elliptic curve is attached to a high-degree rational tail. Notice that the central cone $\sigma_2$ is not smooth; again, we need a Kummer extension (half-lengths).
We also remark that the hyperplane $l_1+l_2=m$ does not come from the alignment, but from the procedure of Definition \ref{dfn:furthersubdivision}. See Figure \ref{esesub3}.
\begin{figure}[h]
\begin{tikzpicture} [scale=.35]
\tikzstyle{every node}=[font=\normalsize]
		\tikzset{arrow/.style={latex-latex}}
		
	\tikzset{cross/.style={cross out, draw, thick,
         minimum size=2*(#1-\pgflinewidth), 
         inner sep=1.2pt, outer sep=1.2pt}}
         
         
        
         
         
        
		\coordinate (E1) at (-1.5,3,0);
		\coordinate (E2) at (1.5,3,0);
		
		\coordinate (T1) at (-3.5,4.5,0);
		\coordinate (bT1) at (-3.5,1.5,0);
		
		\coordinate (T2) at (3.8,4.5,0);
		\coordinate (bT2) at (3.8,1.5,0);
		
	
		\coordinate (E1t) at (-1.5,0,0);
		\coordinate (E2t) at (1.5,0,0);
		\coordinate (T1t) at (-3.5,0,0);
		\coordinate (T2t) at (3.8,0,0);
	
		
		\coordinate (P1) at (-1.8,-1,0);
		\coordinate (P2) at (-1.5,-1.2,0);
		\coordinate (P3) at (-1.2,-1,0);
		\coordinate (P4) at (1.8,-1,0);
		\coordinate (P5) at (1.5,-1.2,0);
    	\coordinate (P6) at (1.2,-1,0);
		
	
		
		\coordinate (l1) at (-1.8,3.3,0);
		\coordinate (bl1) at (-2.5,2.2,0);
		\coordinate (l2) at (1.8,3.3,0);
		\coordinate (bl2) at (2.8,2,0);
		\coordinate (m) at (0,3,0);
		
	
		\coordinate (l1t) at (-2,0.,0);
		\coordinate (l2t) at (2.5,0.,0);
		\coordinate (mt) at (0,0,0);
		
		\begin{scope}[every coordinate/.style={shift={(-6,6,0)}}]
		
		
     
     	
     	\draw ([c]E1)-- ([c]T1);
     	\draw[dotted] ([c]E1)-- ([c]bT1);
     	\draw ([c]E2)-- ([c]T2);
     	\draw[dotted] ([c]E2)-- ([c]bT2);
     	\draw ([c]E1)-- ([c]E2);
     	
     	
     	\draw ([c]E1t)-- ([c]E2t);
     	\draw ([c]E1t)-- ([c]T2t);
     	\draw ([c]E2t)-- ([c]T1t);
     	
     
     	\draw[blue] ([c]E1t)--([c]P1);
     	\draw[blue] ([c]E1t)--([c]P2);
     	\draw[blue] ([c]E1t)--([c]P3);
     	\draw[blue] ([c]E2t)--([c]P4);
     	\draw[blue] ([c]E2t)--([c]P5);
     	\draw[blue] ([c]E2t)--([c]P6);

      
       
       \node at ([c]l1) [above] {\scriptsize{$l_1$}};
        \node at ([c]bl1) [above] {\scriptsize{$l_1$}};
        \node at ([c]l2) [above] {\scriptsize{$l_2$}};
         \node at ([c]bl2) [above] {\scriptsize{$l_2$}};
           \node at ([c]m) [above] {\scriptsize{$m$}};
    
        
      \node at ([c]l1t) [above] {\scriptsize{$l_1$}};
       \node at ([c]l2t) [above] {\scriptsize{$l_2$}};
      \node at ([c]mt) [above] {\scriptsize{$2m$}};
        
       
      \node at ([c]bT1) [left] {\scriptsize{$\overline{T}_1$}};
        \node at ([c]T1) [left] {\scriptsize{$T_1$}};
         
        \node at ([c]bT2) [right] {\scriptsize{$\overline{T}_2$}};
        \node at ([c]T2) [right] {\scriptsize{$T_2$}};
         
         
       
      
       \draw[dashed] ([c]E1)--([c]E1t);
       \draw[dashed] ([c]T1)--([c]T1t);
       \draw[dashed] ([c]T2)--([c]T2t);
       \draw[dashed] ([c]E2)--([c]E2t);
      
     
		\draw[fill=white]  ([c]E1) circle (3pt);
			\draw[fill=white]  ([c]E2) circle (3pt);
			\draw[fill=white]   ([c]E1t) circle (3pt);
			\draw[fill=white]   ([c]E2t) circle (3pt);
			\draw  ([c]bT1) node [cross] {};
			\draw  ([c]bT2) node [cross] {};
			
			\foreach \x in {T1,T2,T1t,T2t} 
          \fill ([c]\x) circle (3pt);
      
         \end{scope}
        
			
      
         \coordinate (A) at (-5,0,0);
         \coordinate (A1) at (-2.5,0,0);
         \coordinate (A2) at (0,0,0);
        \coordinate (A3) at (2.5,0,0);
        \coordinate (B) at (5,0,0);
         \coordinate (B1) at (2.5,4.3,0);
         \coordinate (C) at (0,8.6,0);
         \coordinate (C1) at (-2.5,4.3,0);
         
        
        
         \coordinate (s1) at (0,5,0);
          \coordinate (s2) at (0,2,0);
           \coordinate (s3) at (-1.25,0.5,0);
            \coordinate (s4) at (-3.75,0.5,0);
         
         
         \begin{scope} [every coordinate/.style={shift={(-6,-10,0)}}]
       
        
        \draw ([c]A)--([c]B);
         \draw ([c]A)--([c]C);
          \draw ([c]C)--([c]B);
          
         \draw[green]([c]C1)--([c]B1); 
         \draw[red]([c]C1)--([c]A2); 
        \draw ([c]B1)--([c]A2); 
        
        \draw[orange]([c]A1)--([c]C1); 
         \draw([c]A3)--([c]B1); 
        
        
        \node at ([c]C) [above] {\scriptsize{$m$}};
        \node at ([c]A) [left] {\scriptsize{$l_1$}};
        \node at ([c]B) [right] {\scriptsize{$l_2$}};
        
        \node at ([c]B1) [right] {\begin{color}{green}\scriptsize{$\tau_1$}\end{color}};
         \node at ([c]A1) [below] {\begin{color}{orange}\scriptsize{$\tau_3$}\end{color}};
          \node at ([c]A2) [below] {\begin{color}{red}\scriptsize{$\tau_2$}\end{color}};
        
        
          \node at ([c]s1) [above] {\scriptsize{$\sigma_1$}};
          \node at ([c]s2) [above] {\scriptsize{$\sigma_2$}};
          \node at ([c]s3) [above] {\scriptsize{$\sigma_3$}};
          \node at ([c]s4) [above] {\scriptsize{$\sigma_4$}};
        
         \end{scope}
      
     
      
     
       \coordinate (label)  at (-4,1,0);
      \coordinate (o1) at  (0,1,0);
       \coordinate (t1m) at  (-1,-1.5,0);
       \coordinate (t1c) at  (-1.8,-1.5,0);
        \coordinate (t1p) at  (1,-1.5,0);
        \coordinate (t2m) at  (2,-1.5,0);
         \coordinate (o2) at  (3,1.5,0);
         \coordinate (t2p) at  (4,-1.5,0);
         \coordinate (t2c) at  (4.8,-1.5,0);
        
        
         \coordinate (mm) at (1.5,-1.5,0);
         
        
           \coordinate (nn) at (1.5,-.75,0);
           \coordinate (mn) at (1.5,1.25,0);
        
        
          \coordinate (o1bis) at  (0,2.5,0);
            \coordinate (o2bis) at  (1.5,.5,0);
            \coordinate (t2bis) at  (3,-1.5,0);
          
           
              \coordinate (t2tris) at  (1.7,-.5,0);
               \coordinate (o2tris) at  (1.5,.85,0);
               
               
                
                \coordinate (t2rip) at  (1.7,-1.5,0);
                
           
                 \coordinate (t1mbis) at  (-2.5,-1.5,0);
                  \coordinate (t1mtris) at  (-3.7,-1.5,0);
        
         
         \coordinate (s1) at (-.5,-.35,0);
          \coordinate (s2) at (.4,-.35,0);
          \coordinate (s3) at (2.6,-.15,0);
          \coordinate (s4) at (3.5,-.15,0);
        
         
          
          \coordinate (ss1) at (-.75,0,0);
          \coordinate (ss2) at (.8,1.3,0);
          \coordinate (ss3) at  (2.35,-1.15,0);
           \coordinate (ss4) at  (1.25,.-.5,0);
        
          
            \coordinate (x1) at  (-2.5,-2.5,0);
         \coordinate (x2) at  (.5,.5,0);
         \coordinate(oo1) at (-1.5,-1.5,0);
         \coordinate (m1) at  (-.85,.85,0);
         \coordinate (m2) at  (-3.5,-.5,0);
          \coordinate(oo2) at (1.5,-1.5,0);
         \coordinate (n1) at  (.85,.85,0);
         \coordinate (n2) at  (3.5,-.5,0);
         \coordinate (y1) at  (2.5,-2.5,0);
         \coordinate (y2) at  (-.5,.5,0);
         
          
           \coordinate (z1) at (-3,-1,0);
            \coordinate (z2) at (3,-1,0);
            \coordinate (l1) at (-2,1,0);
            \coordinate (ol) at (-1,-1,0);
            \coordinate (l2) at (0,1,0);
            \coordinate (j1) at (2,-3,0);
            \coordinate (oj) at (1,-1,0);
            \coordinate (j2) at (0,-3,0);
            
         
            \coordinate (z1e) at (-3,-1.2,0);
            \coordinate (z2e) at (3,-1.2,0);
             \coordinate (x1e) at (-2.5,-1.5,0);
            \coordinate (x2e) at (-3,1.2,0);
             \coordinate (y1e) at (2.5,-1.5,0);
            \coordinate (y2e) at (3,1.2,0);
         
           
            \coordinate (g1) at (-2,1,0);
            \coordinate (on) at (0,-1,0);
            \coordinate (g2) at (2,1,0);
             \coordinate (v1) at (0,2,0);
            \coordinate (v2) at (0,-3,0);
            
                 
                  \coordinate (w1) at (-.5,1.7,0);
            \coordinate (w2) at (2.4,2,0);
         
        
           \begin{scope} [every coordinate/.style={shift={(8,14,0)}}]
           \node at ([c]label) {$\sigma_1:$};
          
            \draw ([c]o1)--([c]t1m);
             \draw ([c]o1)--([c]t1p);
            \draw ([c]o2)--([c]t2p);
            \draw ([c]o2)--([c]t2m);
              \draw ([c]t1p)--([c]t2m);
              \draw ([c]t1c)--([c]t1m);
              \draw ([c]t2c)--([c]t2p);
               
              
               
               \node at ([c]s1)  [left] {\begin{color}{blue}\scriptsize{$1$}\end{color}};
             \node at ([c]s2) [right]  {\begin{color}{blue}\scriptsize{$1$}\end{color}};
             \node at ([c]s3)  [left] {\begin{color}{blue}\scriptsize{$1$}\end{color}};
             \node at ([c]s4) [right]  {\begin{color}{blue}\scriptsize{$1$}\end{color}};
               
          
            \draw[fill=white] ([c]o1) circle (3pt);
           \draw[fill=white] ([c]o2) circle (3pt);
             \fill ([c]t2p) circle (3pt);
              \fill ([c]t1m) circle (3pt);
              
              
              \draw ([c]t2m) node[cross,red]{};
            \draw ([c]t1p) node[cross,red]{};
            
           \end{scope}
            \begin{scope} [every coordinate/.style={shift={(19,14,0)}}]
            \draw[gray] ([c]x1)--([c]x2);
                \draw[gray] ([c]y1)--([c]y2);
                \draw[thick] ([c]oo1) to[out=45,in=270] ([c]m1);
               \draw[thick] ([c]oo1) to[out=225,in=270] ([c]m2); 
                \draw[thick] ([c]oo2) to[out=135,in=270] ([c]n1);
               \draw[thick] ([c]oo2) to[out=-45,in=270] ([c]n2);
               
                          \node at ([c]m2)  [left] {\scriptsize{$T_1$}};;
                        \node at ([c]n2) [right]   {\scriptsize{$T_2$}};
        \end{scope}
           
           
         \begin{scope} [every coordinate/.style={shift={(8,8,0)}}]
           \node at ([c]label) {\begin{color}{green}$\tau_1:$\end{color}};
        
            
            \draw ([c]o1)--([c]mm);
             \draw ([c]o1)--([c]t1m);
            \draw ([c]o2)--([c]t2p);
            \draw ([c]o2)--([c]mm);
             \draw ([c]t1c)--([c]t1m);
              \draw ([c]t2c)--([c]t2p);
               
              
               
               \node at ([c]s1)  [left] {\begin{color}{blue}\scriptsize{$1$}\end{color}};
             \node at ([c]s2) [below]  {\begin{color}{blue}\scriptsize{$1$}\end{color}};
             \node at ([c]s3)  [below] {\begin{color}{blue}\scriptsize{$1$}\end{color}};
             \node at ([c]s4) [right]  {\begin{color}{blue}\scriptsize{$1$}\end{color}};
               
              
                  \draw[fill=white] ([c]o1) circle (3pt);
           \draw[fill=white] ([c]o2) circle (3pt);
             \fill ([c]t2p) circle (3pt);
              \fill ([c]t1m) circle (3pt);
              
              
              \draw ([c]mm) node[cross,red]{};
            
               
           \end{scope}
           
           
        
         \begin{scope} [every coordinate/.style={shift={(19,8,0)}}]
            \draw[gray] ([c]z1)--([c]z2);
              
                \draw[thick] ([c]ol) parabola ([c]l1);
               \draw[thick] ([c]ol) parabola ([c]l2); 
                \draw[thick] ([c]oj) parabola ([c]j1);
               \draw[thick] ([c]oj) parabola ([c]j2);
               
                          \node at ([c]l2)  [left] {\scriptsize{$T_1$}};
                        \node at ([c]j2) [right]   {\scriptsize{$T_2$}};
        \end{scope}
        
         \begin{scope} [every coordinate/.style={shift={(8,2,0)}}]
           \node at ([c]label) {$\sigma_2:$};
           
            
            
            \draw ([c]o1)--([c]nn);
             \draw ([c]o1)--([c]t1m);
            \draw ([c]o2)--([c]t2p);
            \draw ([c]o2)--([c]nn);
             \draw ([c]t1c)--([c]t1m);
              \draw ([c]t2c)--([c]t2p);
               
              
               
               \node at ([c]s1)  [left] {\begin{color}{blue}\scriptsize{$1$}\end{color}};
             \node at ([c]s2)  {\begin{color}{blue}\scriptsize{$1$}\end{color}};
             \node at ([c]s3) {\begin{color}{blue}\scriptsize{$1$}\end{color}};
             \node at ([c]s4) [right]  {\begin{color}{blue}\scriptsize{$1$}\end{color}};
             
              \draw [decorate,decoration={brace,amplitude=5pt,mirror},xshift=0.2pt,yshift=-0.4pt] ([c]nn)--([c]o1) node[black,midway,xshift=-0.5cm,yshift=0.6cm] {\scriptsize{$\frac{l_1+m-l_2}{2}$}};
           \draw [decorate,decoration={brace,amplitude=5pt,mirror},xshift=0.2pt,yshift=-0.3pt] ([c]o2)--([c]nn) node[black,midway,yshift=0.6cm] {\scriptsize{$\frac{l_2+m-l_1}{2}$}};
               
              
               \draw[fill=white] ([c]o1) circle (3pt);
           \draw[fill=white] ([c]o2) circle (3pt);
             \fill ([c]t2p) circle (3pt);
              \fill ([c]t1m) circle (3pt);
              
              
              \draw ([c]nn) node[cross,red]{};
               
           \end{scope}
           
           
        
         \begin{scope} [every coordinate/.style={shift={(19,2,0)}}]
            \draw[gray] ([c]z1)--([c]z2)--([c]z2e)--([c]z1e)--([c]z1);
             \draw[thick] ([c]x1e)--([c]x2e); 
              \draw[thick] ([c]y1e)--([c]y2e);  
               
                          \node at ([c]x2e)  [left] {\scriptsize{$T_1$}};
                        \node at ([c]y2e) [right]   {\scriptsize{$T_2$}};
        \end{scope}
        
       
        \begin{scope} [every coordinate/.style={shift={(8,-4,0)}}]
           \node at ([c]label) {\begin{color}{red}$\tau_2:$\end{color}};
          
              
            
            
            \draw ([c]o1bis)--([c]o2bis);
             \draw ([c]o1bis)--([c]t1m);
            \draw ([c]o2bis)--([c]t2bis);
            \draw ([c]t2bis)--([c]t2c);
             \draw ([c]t1c)--([c]t1m);
             
               
              
               
               \node at ([c]ss1)  [right] {\begin{color}{blue}\scriptsize{$1$}\end{color}};
             \node at ([c]ss2) [above]  {\begin{color}{blue}\scriptsize{$1$}\end{color}};
             \node at ([c]ss3)   {\begin{color}{blue}\scriptsize{$1$}\end{color}};
            
             
               \draw[fill=red] ([c]o2bis) circle (3pt);
           \draw[fill=white] ([c]o1bis) circle (3pt);
             \fill ([c]t2bis) circle (3pt);
              \fill ([c]t1m) circle (3pt);
              
              
               
               
           \end{scope}
        
       
        \begin{scope} [every coordinate/.style={shift={(19,-4,0)}}]
            \draw[gray] ([c]z1)--([c]z2)--([c]z2e)--([c]z1e)--([c]z1);
             \draw[thick] ([c]x1e)--([c]x2e); 
              \draw[thick] ([c]y1e)--([c]y2e);  
               
                          \node at ([c]x2e)  [left] {\scriptsize{$T_1$}};
                        \node at ([c]y2e) [right]   {\scriptsize{$T_2$}};
        \end{scope}
        
         \begin{scope} [every coordinate/.style={shift={(8,-10,0)}}]
           \node at ([c]label) {$\sigma_3:$};
          
            
            
            \draw ([c]o1bis)--([c]o2tris);
             \draw ([c]o1bis)--([c]t1m);
            \draw ([c]o2tris)--([c]t2tris);
             \draw ([c]t2bis)--([c]t2tris);
            \draw ([c]t2bis)--([c]t2c);
             \draw ([c]t1c)--([c]t1m);
             
               
              
               
               \node at ([c]ss1)  [right] {\begin{color}{blue}\scriptsize{$1$}\end{color}};
             \node at ([c]ss2) [above]  {\begin{color}{blue}\scriptsize{$1$}\end{color}};
             \node at ([c]ss3)  [above] {\begin{color}{blue}\scriptsize{$1$}\end{color}};
             \node at ([c]ss4)  [above] {\begin{color}{blue}\scriptsize{$2$}\end{color}};
            
              
               \draw [decorate,decoration={brace,amplitude=5pt,mirror},xshift=0.2pt,yshift=-0.2pt] ([c]t2tris)--([c]o2tris) node[black,midway,xshift=0.8cm,yshift=0.2cm] {\scriptsize{$l_1-m-l_2$}};
           \draw [decorate,decoration={brace,amplitude=5pt,mirror},xshift=0.2pt,yshift=-0.5pt] ([c]t2tris)--([c]t2bis) node[black,midway,xshift=-0.2cm,yshift=-0.4cm] {\scriptsize{$m+2l_2-l_1$}};

            
              \draw[fill=white]([c]o1bis) circle (3pt);
           \draw[fill=white] ([c]o2tris) circle (3pt);
             \fill ([c]t2bis) circle (3pt);
              \fill ([c]t1m) circle (3pt);
              
               \draw ([c]t2tris) node[cross,red]{};
              
             
             
           \end{scope}
        
       
        \begin{scope} [every coordinate/.style={shift={(19,-10,0)}}]
            \draw[gray] ([c]z1)--([c]z2)--([c]z2e)--([c]z1e)--([c]z1);
             \draw[thick] ([c]x1e)--([c]x2e); 
              \draw[thick] ([c]y1e)--([c]y2e);  
               
                          \node at ([c]x2e)  [left] {\scriptsize{$T_1$}};
                        \node at ([c]y2e) [right]   {\scriptsize{$T_2$}};
        \end{scope}
 
  \begin{scope} [every coordinate/.style={shift={(8,-16,0)}}]
           \node at ([c]label) {\begin{color}{orange}$\tau_3:$\end{color}};
          
              
               
              
            
            
            \draw ([c]o1bis)--([c]o2tris);
             \draw ([c]o1bis)--([c]t1m);
            \draw ([c]o2tris)--([c]t2rip);
           
            \draw ([c]t2rip)--([c]t2c);
             \draw ([c]t1c)--([c]t1m);
             
               
              
               
               \node at ([c]ss1)  [right] {\begin{color}{blue}\scriptsize{$1$}\end{color}};
             \node at ([c]ss2) [above]  {\begin{color}{blue}\scriptsize{$1$}\end{color}};
             
             \node at ([c]ss4)  [above] {\begin{color}{blue}\scriptsize{$2$}\end{color}};
            
           
             \draw[fill=white]([c]o1bis) circle (3pt);
           \draw[fill=white] ([c]o2tris) circle (3pt);
             \draw[fill=red] ([c]t2rip) circle (3pt);
              \fill ([c]t1m) circle (3pt);
              
               \end{scope}
               
               
               \begin{scope} [every coordinate/.style={shift={(19,-16,0)}}]
            \draw[gray] ([c]z1)--([c]z2);
              
                \draw[thick] ([c]on) parabola ([c]g1);
               \draw[thick] ([c]on) parabola ([c]g2); 
                \draw[thick] ([c]v1) -- ([c]v2);
              
               
                          \node at ([c]v1)  [left] {\scriptsize{$T_1$}};
                        \node at ([c]g2) [right]   {\scriptsize{$T_2$}};
        \end{scope}
        
        \begin{scope} [every coordinate/.style={shift={(8,-22,0)}}]
           \node at ([c]label) {$\sigma_4:$};
          
              
            
            
            \draw ([c]o1bis)--([c]o2tris);
             \draw ([c]o1bis)--([c]t1m);
            \draw ([c]o2tris)--([c]t2rip);
           
            \draw ([c]t2rip)--([c]t2c);
             \draw ([c]t1mtris)--([c]t1mbis);
             \draw ([c]t1mbis)--([c]t1m);
               
              
               
               \node at ([c]ss1)  [right] {\begin{color}{blue}\scriptsize{$1$}\end{color}};
             \node at ([c]ss2) [above]  {\begin{color}{blue}\scriptsize{$1$}\end{color}};
             
             \node at ([c]ss4)  [above] {\begin{color}{blue}\scriptsize{$2$}\end{color}};
            
             
               \draw[fill=white]([c]o1bis) circle (3pt);
           \draw[fill=white] ([c]o2tris) circle (3pt);
             \draw[fill=red] ([c]t2rip) circle (3pt);
              \fill ([c]t1mbis) circle (3pt);
              
                 \draw ([c]t1m) node[cross]{};
                 
               \end{scope}
               
               
               \begin{scope} [every coordinate/.style={shift={(19,-22,0)}}]
            \draw[gray] ([c]z1)--([c]z2);
              
                \draw[thick] ([c]on) parabola ([c]g1);
               \draw[thick] ([c]on) parabola ([c]g2); 
                \draw[gray] ([c]v1) -- ([c]v2);
                \draw[thick] ([c]w1) -- ([c]w2);
              
               
                          \node at ([c]w1)  [left] {\scriptsize{$T_1$}};
                        \node at ([c]g2) [right]   {\scriptsize{$T_2$}};
        \end{scope}
        
  
        
  \end{tikzpicture}
	\caption{Tropical admissible cover (top left); the subdivision of $\operatorname{trop}(\mathcal A_2^{\text{wt}})$ (bottom left);  $\lambda_T$  and the  Gorenstein singularities (right).}
 \label{esesub3}
\end{figure}
\end{exa}

\newpage


\subsection{The main construction}\label{sec:main}
We construct a universal morphism $\widetilde{\mathcal C}\to \overline{\mathcal C}$ over $\widetilde{\mathcal A}_{2}$, where $\overline{\mathcal C}$ is a family of Gorenstein singularities (both isolated and ribbons). We do so in two steps: first, a contraction informed by $\lambda$, producing a possibly non-Gorenstein curve. The image of the non-Gorenstein locus is contained in a divisor, which we name $\pazocal D_1$ below. We complete the construction of $\overline{\mathcal C}$ by gluing in a portion of $\psi$ over this locus, thus producing a non-reduced structure along the fibres.

\begin{dfn}
 Let $\rho_\text{max}$ denote the maximum value attained by $\lambda$ on $\scalebox{0.8}[1.3]{$\sqsubset$}$, and let $\pazocal D\subseteq\widetilde{\mathcal A}_{2}$ be the Cartier divisor determined by the ideal sheaf $\mathcal O_{\widetilde{\mathcal A}}(-\rho_\text{max})\hookrightarrow\mathcal O_{\widetilde{\mathcal A}}$.
\end{dfn} 
\begin{dfn}
Let $\rho_1$ be the value of $\lambda$ at the vertex supporting $K_{\scalebox{0.8}[1.3]{$\sqsubset$}}+\operatorname{div}(\lambda)$, and let $\pazocal D_1\subseteq\widetilde{\mathcal A}_{2}$ be the Cartier divisor determined by the ideal sheaf $\mathcal O_{\widetilde{\mathcal A}}(-\rho_1)\hookrightarrow\mathcal O_{\widetilde{\mathcal A}}$. 
\end{dfn} Note that $\rho_1\leq\rho_\text{max}$ implies $\pazocal D_1\subseteq \pazocal D$. Moreover, $\pazocal D_1$ has three \emph{connected} components, denoted by $\pazocal D_1^0,\pazocal D_1^1$, and $\pazocal D_1^2$, according to the weight of $\scalebox{0.9}[1.3]{$\triangle$}^\circ$.
\begin{dfn}
 Let $\pazocal D_2$ be cut out by $\rho_\text{max}-\rho_1$, so that we have an exact sequence:
 \[0\to\mathcal O_{\pazocal D_2}(-\rho_1)\to\mathcal O_{\pazocal D}\to\mathcal O_{\pazocal D_1}\to0\]
 \end{dfn}
 \begin{dfn}
 Let $\pazocal Z$ be the Cartier divisor on $\widetilde{\mathcal C}$ determined by the inclusion $\mathcal O_{\widetilde{\mathcal C}}(-\lambda)\hookrightarrow\mathcal O_{\widetilde{\mathcal C}}$. It is supported over $\pazocal D$.
\end{dfn}

We choose smooth disjoint sections $x_1,\ldots,x_{2d}$ of $\widetilde{\mathcal C}$ according to the weight function, namely in a way that $\#\{i|p_i\in \widetilde{C}_v\}=2w(v)$.

\begin{rem}\label{rem:localsections}
  Such sections exist only locally, but we will show that our construction does not depend on this choice, therefore it glues on the whole of $\widetilde{\mathcal A}_2$.
\end{rem}

 
\noindent \emph{Notation.} Let $\sigma\colon\widetilde{\mathcal C}\to\widetilde{\mathcal C}$ denote the hyperelliptic involution. Let $p_i$ denote the multi-section $x_i+\sigma(x_i)$. We denote by $\mathcal L$ the line bundle $\omega_{\widetilde{\mathcal C}/\widetilde{\mathcal A}}(\mathbf{p})(\lambda)$ on $\widetilde{\mathcal C}$.

\begin{thm}\label{thm:contraction}
 Upon contracting tails away from $\scalebox{0.9}[1.3]{$\triangle$}$, $\mathcal L$ is $\pi$-semiample. In the diagram
 \begin{center}\begin{tikzcd}
 \widetilde{\mathcal C}\ar[rr,"\phi"]\ar[dr,"\pi"] & &\mathcal{C}^\prime:=\underline{\operatorname{Proj}}_{\widetilde{\mathcal A}}\left(\pi_*\bigoplus_{k\geq 0}\mathcal L^{\otimes k}\right)\ar[dl,"\pi^\prime"] \\
 & \widetilde{\mathcal A}_{2} &
 \end{tikzcd}\end{center}
 the morphism $\pi^\prime$ is a flat family of reduced, projective, Cohen-Macaulay curves of arithmetic genus two, with Gorenstein fibres outside $\pazocal D_1$. Moreover, we can perform a parallel contraction $\mathcal T\to \mathcal{T}^\prime$ so that $\psi^\prime\colon \mathcal{C}^\prime\to \mathcal{T}^\prime$ remains finite. Neither $\mathcal{C}^\prime$ nor $\mathcal{T}^\prime$ depend on the choice of sections respecting the weight function, as per Remark \ref{rem:localsections}.
\end{thm}

\begin{rem}\label{rem:tailcontraction}
 We have required $T$ to be weighted-stable, but it is still possible that $\widetilde{C}$ contains unmarked rational tails and rational bridges (as conjugate of stable components).
 Once the singularity is formed, all those unstable components that do not map to its special branches can safely be collapsed. On the other hand, it is convenient to keep them through $C^\prime$, so that the latter is hyperelliptic too. These unstable components are harmless for our purposes, since they do not interfere with the factorisation condition.
\end{rem}

\begin{thm}\label{thm:pushout}
 Let $\pazocal{Z}^\prime$ denote the image of $\pazocal Z$ under $\phi$. Over $\pazocal D_1$, it is a flat family of Gorenstein curves of arithmetic genus two, and the image of $\pazocal{Z}^\prime$ under $\psi^\prime$ is a (chain of) rational curve(s) $\mathcal{T}^\prime_{\pazocal Z}$. 
 Let $\overline{\mathcal C}$ be obtained as the pushout of:
 \begin{center}\begin{tikzcd}
 \pazocal{Z}^\prime_{\pazocal D_1}\ar[d,"\psi^\prime"]\ar[r,hook] &\mathcal{C}^\prime \\
 \mathcal{T}^\prime_{\pazocal Z} &
 \end{tikzcd}\end{center}
 Then $\overline{\mathcal C}\to\widetilde{\mathcal A}_{2}$ is a flat family of projective, Gorenstein curves of arithmetic genus two. Moreover, the weight of the minimal subcurve of genus two is at least three.
\end{thm}

\subsection{First step: the contraction}

\begin{lem}\label{lem:basechange}
 For $k\geq2$, $R^1\pi_*\mathcal L^{\otimes k}$ is supported along $\pazocal D_2$; moreover, it admits a two-term resolution that remains such after pullback to a sufficiently generic base: in particular, if $f\colon T\to\widetilde{\mathcal A}_{2}$ is a morphism such that $\mathcal O_T(-\rho_\text{max})\to\mathcal O_T$ remains injective, then
 \[f^*\pi_*(\mathcal L^{\otimes k})\to(\pi_T)_*(\mathcal L_T^{\otimes k})\]
 is an isomorphism.
\end{lem}

\begin{proof}
\emph{Support:} Since $\mathcal{L}^{\otimes k}$ is flat on the base, by cohomology and base change it is enough to show the vanishing of  $H^1(\widetilde{C}_s,L^{\otimes k})$ for $s\in \widetilde{\mathcal A}_2\setminus\pazocal D_2.$

By Remark~\ref{rem:tailcontraction} we may assume that $\mathcal L$ is relatively ample outside of $\pazocal Z$, and in particular over $\widetilde{\mathcal A}_2\setminus \pazocal D$.

For $s\in \pazocal D\setminus\pazocal D_2$ we have $\rho_1=\rho_{\max}$.
We argue that $L\geq0$ and it has degree two on every positive genus subcurve of $\widetilde{C}_s$. Then, it is clear by degree reasons that $h^1(L^{\otimes k})=0$ for $k\geq 2$.

It is easier to argue on $\scalebox{0.9}[1.2]{$\top$}$. We will say that a vertex $w_T$ has positive genus if it corresponds to a vertex $w$ of $\scalebox{0.8}[1.3]{$\sqsubset$}$ of genus one or two. $T$ is marked with the branch locus of $\psi$, and correspondingly $\scalebox{0.9}[1.2]{$\top$}$ has six unmarked infinite legs (we call them $B$-legs). The Riemann-Hurwitz formula implies that:
\begin{equation}
 g(w)=\frac{1}{2}(\#\{f\ B\text{-leg }| f\multimap w_T\}+\#\{e\multimap w_T| s(\lambda,e)\in\frac{1}{2}\mathbb Z\}-2).
\end{equation}

By definition of admissible function, a vertex of $\partial\scalebox{0.9}[1.3]{$\triangle$}$ can have positive genus only if it is the vertex supporting $D$ (see Lemma \ref{lem:highergenusonbdry}). Thus, outside $\pazocal D_2$, $v\in\partial\scalebox{0.9}[1.3]{$\triangle$}$ is rational and supports at most two $B$-legs; we claim that it supports at most one. Indeed, if $e$ is the unique edge from $v$ to $\scalebox{0.9}[1.3]{$\triangle$}^\circ$, by rewriting \eqref{eqn:Dv} we would find:
\[-s(e,\lambda_T)=\operatorname{val}(v)-2+\frac{1}{2}\#\{f\ B\text{-leg }| f\multimap v\}=0\]
if there were two $B$-legs, so $v$ would not lie on the boundary. It follows that $s(e,\lambda_T)=1$ if there are no $B$-legs on $v$, and $s(e,\lambda_T)=\frac{1}{2}$ if there is one.

Now we claim that every vertex $w$ such that $0<\lambda(w)<\rho_\text{max}$ is rational. We prove it by induction on the level, or distance from $\partial\scalebox{0.9}[1.3]{$\triangle$}$. The claim reduces to the following: a vertex of positive genus with an outgoing edge on which $\lambda$ has positive slope must be the vertex carrying $D$. This cannot happen when $\rho_1=\rho_\text{max}$.

By rewriting \eqref{eqn:Dv} as:
\[D(w_T)=\sum_{e\multimap w_T}(s(\lambda_T,e)+1)-1+\frac{1}{2}(\#\{f\ B\text{-leg }| f\multimap w_T\}-2),\]
and by the inductive assumption that all the downward slopes are either $-1$ or $-\frac{1}{2}$, we conclude that $D(w_T)\geq g(w)$.

Finally, by the same formulae, it is easy to see that the top level can be either of the ones represented in Figure \ref{top level D1}.



\begin{figure}[htb]
	\centering


		\begin{tikzpicture} [scale=.6]
		\tikzstyle{every node}=[font=\normalsize]
		\tikzset{arrow/.style={latex-latex}}
\coordinate (A) at (0,3,0);
  
 
 \coordinate (B1) at (-2,3,0);
 \coordinate (B2) at (2,3,0);
  
   
  \coordinate (A1) at (0,3.3,0);
 \coordinate (A2) at (0,2.7,0);
  \coordinate (B3) at (0,3.3,4);  
    
\fill[red] (A) circle (3pt);
\node at (A) [below] {\scriptsize{$g=2$}};


\begin{scope}[every coordinate/.style={shift={(0,-3,0)}}]
\draw ([c]B1) to [out=80,in=100] ([c]A);
\draw ([c]B1) to [out=-80,in=-100] ([c]A);

\draw ([c]A) to [out=80,in=100] ([c]B2);
\draw ([c]A) to [out=-80,in=-100] ([c]B2);
  
\fill[red] ([c]A) circle (3pt);
\fill ([c]B1) circle (3pt);
\fill ([c]B2) circle (3pt);
\end{scope}

\begin{scope}[every coordinate/.style={shift={(5,0,0)}}]
\draw ([c]A) to [out=80,in=100] ([c]B2);
\draw ([c]A) to [out=-80,in=-100] ([c]B2);
  
\fill[red] ([c]A) circle (3pt);
\fill ([c]B2) circle (3pt);
\node at ([c]A) [below,left] {\scriptsize{$g=1$}};
\end{scope}

\begin{scope}[every coordinate/.style={shift={(7,-3,0)}}]
\draw ([c]B1) to [out=80,in=120] ([c]A1);
\draw ([c]B1) to [out=-80,in=-120] ([c]A2);

\draw ([c]A1) to [out=60,in=100] ([c]B2);
\draw ([c]A2) to [out=-60,in=-100] ([c]B2);
 
 \draw ([c]B3) to [out=120,in=100] ([c]A1);
\draw ([c]B3) to [out=-80,in=-100] ([c]A2);   
  
\fill[red] ([c]A1) circle (3pt);
  \fill[red] ([c]A2) circle (3pt);
 
 \foreach \x in {B1,B2,B3} 
\fill ([c]\x) circle (3pt);

\end{scope}

\end{tikzpicture}
\caption{The top level of $\Delta$ over $\pazocal D_1\setminus\pazocal D_2$.}
\label{top level D1}
\end{figure}





\noindent\emph{Resolution:}
Note first that the rank of $R^1\pi_*\mathcal L^{\otimes k}$ is not constant along $\pazocal D_2$.

Indeed, $\pazocal D_2$ has two types of irreducible components:
\begin{itemize}[leftmargin=.5cm]
 \item $\pazocal D_{2,1}$, where \emph{generically} $\scalebox{0.9}[1.3]{$\triangle$}^\circ$ has genus one, and $R^1\pi_*\mathcal L^{\otimes k}$ has rank one;
 \item $\pazocal D_{2,2}$, where $\scalebox{0.9}[1.3]{$\triangle$}^\circ$ has genus two, and $R^1\pi_*\mathcal L^{\otimes k}$ has rank two.
\end{itemize}
Away from their intersection, it is easy to find the desired \emph{local} resolution:
\[0\to\mathcal{O}_U^{\oplus 2}\xrightarrow{\left(\substack{e^{\rho_{\max}}\;\; 0\\ 0\;\;e^{\rho_{\max}}}\right)}\mathcal{O}_U^{\oplus 2}\to \mathcal{O}_{\pazocal D_{2,2}\cap U}^{\oplus 2}\to 0\]
on $U\subseteq\widetilde{\mathcal A}_2$ a neighbourhood of a generic point of $\pazocal D_{2,2}$ (assuming $\rho_1=0$); and 
\[0\to\mathcal{O}_U\xrightarrow{e^{\rho_{\max}}}\mathcal{O}_U\to \mathcal{O}_{\pazocal D_{2,1}\cap U}\to 0\]
around a generic point in $\pazocal D_{2,1}.$

The intersection of $\pazocal D_{2,1}$ with $\pazocal D_{2,2}$ has two types of irreducible components, see Figure \ref{intersection with D2}. In order to obtain a local resolution, we adapt an argument of \cite{HLN}. We sketch it here for the reader's benefit. 

\begin{figure}
	\centering


		\begin{tikzpicture} [scale=.6]
		\tikzstyle{every node}=[font=\normalsize]
		\tikzset{arrow/.style={latex-latex}}

\coordinate (E1) at (-1,4,0);

\coordinate (E1l) at (-1,4.2,0);
\coordinate (E2) at (2,2.5,0);

\coordinate (E12) at (0.5,3.25,0);
\coordinate (E2l) at (2,2.7,0);

\coordinate (E2W) at (2.75,1.25);
\coordinate (W) at (3.5,0,0);

\coordinate (A1) at (-2.5,4.3,0);
\coordinate (B1) at (-2.5,3.7,0);

\coordinate (A2) at (.5,2.8,0);
\coordinate (B2) at (.5,2.2,0);

\coordinate (T1) at (-2.5,0,0);
\coordinate (T2) at (-2,0,0);
\coordinate (T3) at (-1.5,0,0);
\coordinate (T4) at (0.7,0,0);
\coordinate (T5) at (1.3,0,0);

\draw[->-] (W) to (E2);
\draw[->-] (E2) to (E1);

\draw (T4) to (E2);
\draw (T5) to (E2);

\draw (T1) to (E1);
\draw (T2) to (E1);
\draw (T3) to (E1);

\draw[gray] (A1) to (E1);
\draw[gray] (B1) to (E1);
\draw[gray] (B2) to (E2);
\draw[dashed] (-3,0,0)--(4,0,0);
\foreach \x in {T1,T2,T3,T4,T5} 
\fill (\x) circle (3pt);
\fill[red] (W) circle (3pt);

\draw[fill=white] (E1) circle (3.5pt);
\draw[fill=white]  (E2) circle (3.5pt);


\node at (E1l) [above] {\scriptsize{$E_1$}};
\node at (E1l) [right] {\scriptsize{$g=1$}};
\node at (E2l) [above] {\scriptsize{$E_2$}};
\node at (E2l) [right] {\scriptsize{$g=1$}};

\node at (E12) [above] {\color{blue}\scriptsize{$1$}};
\node at (E12) [below] {\scriptsize{$\zeta_1$}};

\node at (E2W) [right] {\color{blue}\scriptsize{$2$ or $3$}};
\node at (E2W) [left] {\scriptsize{$\zeta_2$}};

\node at (A1) [left] {\begin{color}{gray}\scriptsize{$A_1$}\end{color}};
\node at (B1) [left] {\begin{color}{gray}\scriptsize{$B_1$}\end{color}};
\node at (B2) [left] {\begin{color}{gray}\scriptsize{$A_2$}\end{color}};

\node at (T3) [above right] {\color{blue}\scriptsize{$1$}};
\node at (T5) [above right] {\color{blue}\scriptsize{$1$}};

\node at (0.5,-.5,0) [below]  {\scriptsize{$\pazocal D^{EE}$}};



\begin{scope}[every coordinate/.style={shift={(10,0,0)}}]


\draw[->-] ([c]W) to ([c]E2);
\draw[->-] ([c]E2)  to [out=80,in=80]([c]E1);
\draw[->-] ([c]E2)  to [out=-100,in=-100]([c]E1);

\draw ([c]T4) to ([c]E2);
\draw ([c]T5) to ([c]E2);

\draw ([c]T1) to ([c]E1);
\draw ([c]T2) to ([c]E1);
\draw ([c]T3) to ([c]E1);

\draw[gray] ([c]A1) to ([c]E1);
\draw[gray] ([c]B1) to ([c]E1);
\draw[gray] ([c]A2) to ([c]E2);

\draw[dashed] (7,0,0)--(14,0,0);
\foreach \x in {E2,T1,T2,T3,T4,T5} 
\fill ([c]\x) circle (3pt);
\fill[red] ([c]W) circle (3pt);

\draw[fill=white] ([c]E1) circle (3.5pt);



\node at (8.9,4.1,0) [above] {\scriptsize{$E_1$}};
\node at (9,4,0) [right] {\scriptsize{$g=1$}};
\node at([c]E2) [right] {\scriptsize{$E_2$}};


\node at (11,4,0) [above] {\color{blue}\scriptsize{$1$}};
\node at (10.25,3,0)  [below] {\color{blue}\scriptsize{$1$}};
\node at (10.5,2.35,0) [below] {\scriptsize{$\zeta_1$}};

\node at ([c]E2W) [right] {\color{blue}\scriptsize{$2$ or $3$}};
\node at ([c]E2W) [left] {\scriptsize{$\zeta_2$}};

\node at ([c]A1) [left] {\begin{color}{gray}\scriptsize{$A_1$}\end{color}};
\node at ([c]B1) [left] {\begin{color}{gray}\scriptsize{$B_1$}\end{color}};
\node at ([c]A2) [above left] {\begin{color}{gray}\scriptsize{$A_2$}\end{color}};

\node at ([c]T3) [above right] {\color{blue}\scriptsize{$1$}};
\node at ([c]T5) [above right] {\color{blue}\scriptsize{$1$}};

\node at (10.5,-.5,0) [below]  {\scriptsize{$\pazocal D^{B}$}};


\end{scope}



\end{tikzpicture}






 \caption{The generic points of $\pazocal D_{2,1}\cap\pazocal D_{2,2}$.}
 \label{intersection with D2}
\end{figure}




Locally on the base, there is a section $x_{2d+1}$ such that, setting $p_{2d+1}=x_{2d+1}+\sigma(x_{2d+1})$, we have $\omega_C(\lambda)=\mathcal O_C(p_{2d+1})$, and therefore
$\mathcal L=\mathcal O_C(\sum_{i=1}^{2d+1} p_i)$.

Again locally on the base, we may find disjoint generic sections $A_1$, $A_2$, and $B$, such that $A_1$ and $B$ pass through $E_1$, and $A_2$ passes through $E_2$. Then $\mathcal L(A_1+A_2-B)$ has vanishing $h^1$ on fibres, and therefore, by cohomology and base-change, $\pi_*\mathcal L(A_1+A_2-B)$ is a vector bundle. Locally, $\pi_*\mathcal L$ differs from $\pi_*\mathcal L(-B)$ by a trivial line bundle, and the latter is the kernel of the evaluation map:
\[\pi_*\mathcal L(A_1+A_2-B)\to\pi_*(\mathcal O_{A_1}(A_1)\oplus\mathcal O_{A_2}(A_2)).\]
The evaluation map can be studied fibrewise, since both sheaves in question have vanishing $h^1$; moreover, the source can be decomposed into:
\[\pi_*\mathcal L(A_1+A_2-B)\cong\bigoplus_{i=1}^{2d+2}\pi_*\mathcal O(p_i+A_1+A_2-B),\]
and the evaluation map can be studied componentwise:
\[\operatorname{ev}_{i,j}\colon \pi_*\mathcal O(p_i+A_1+A_2-B)\to\pi_*(\mathcal O_{A_j}(A_j))\quad i=1,\ldots,2d+2;j=1,2. \]
The cokernel of the latter is $H^1(C,\mathcal O_C(p_i+A_{2-j}-B))$. See \cite[\S 4.2]{HL}.

It is not hard to see that the latter is non-zero precisely when $p_i$ stays away from $E_j$, therefore, in some local trivialisation of the line bundles involved,
\[\operatorname{ev}_{i,j}=c_{i,j}\prod_{q\in[p_i,A_j]}\zeta_q\]
where $c_{i,j}\in\mathcal O_{U}^*$, $[p_i,A_j]$ denotes the set of nodes separating $p_i$ from $A_j$, and $\zeta_q\in\mathcal O_{U}$ is the smoothing parameter of the node $q$. Thanks to the alignment,
\begin{itemize}[leftmargin=.5cm]
 \item the smoothing parameter $\zeta_2$ of the node separating $E_2$ from the component supporting $D$ divides all the expressions of the form $\operatorname{ev}_{i,2},\ i=1,\ldots,2d+2$;
 \item if $\zeta_1$ denotes the smoothing parameter of the node separating $E_1$ from $E_2$, the product $\zeta_1\zeta_2$ divides all the expressions of the form $\operatorname{ev}_{i,1},\ i=1,\ldots,2d+2$.
\end{itemize}
We can therefore use the column of the evaluation matrix associated to a marking $p_i$ (up to relabelling, $i=1$) on the component supporting $D$ in order to put the matrix in triangular form. In order to diagonalise it, we need a more refined information that we borrow from \cite[\S 2.6-7]{HLN}, and we restate here in streamlined form:

\smallskip

\noindent\textbf{Proposition}[Hu-Li-Niu] \emph{The determinant of the matrix:}
\[\begin{pmatrix}
   c_{1,1} & c_{j,1} \\
   c_{1,2} & c_{j,2}
  \end{pmatrix}\]
\emph{is invertible when the markings $p_1$ and $p_j$ are not conjugate under $\psi$. }

\smallskip

Since the component supporting $D$ contains at least three markings ($p_1,p_{2d+1}$, and $p_{2d+2}$), we can find two non-conjugate ones.

Summing up, the evaluation matrix can be put in the following form:
\[\begin{pmatrix}
   \zeta_1 & 0 \\
   0 & \zeta_1\zeta_2
  \end{pmatrix}\]
(the remaining columns are zero). Noticing that $\pazocal D_{2,i}=\{\zeta_i=0\},\ i=1,2$, around the given point, we have thus found the desired local resolution of $\operatorname{R}^1\pi_*\mathcal L^k$. In particular, it follows that $\pi_*\mathcal L^{\otimes\geq2}$ is a vector bundle on $\widetilde{\mathcal A}_2$.

\medskip

\noindent\emph{Base-change:} see the proof of \cite[Lemma 3.7.5.3]{RSPW1}. Note that the evaluation matrix above is an explicit instance of the Grothendieck-Mumford's complex for cohomology and base-change, and $\zeta_1\zeta_2=e^{\rho_{\text{max}}}$.

\end{proof}


\begin{proof}[Proof of Theorem~\ref{thm:contraction}] Analogous to \cite[Lemma 2.13]{SMY1} and \cite[Proposition 3.7.6.1]{RSPW1}. We recap for the reader's convenience.

\noindent\emph{Flatness}: is equivalent to $\pi_*\mathcal L^{\otimes k}$ being a vector bundle for $k\gg0$ \cite[\href{https://stacks.math.columbia.edu/tag/0D4D}{Tag 0D4D}]{stacks-project}.

\noindent\emph{Basepoint freeness}: i.e. existence of the morphism $\phi$. This is clear outside $\pazocal D$ where  $\mathcal{L}^{\otimes k}$ is $\pi-$ ample. Even on $\pazocal D$ we have that $\mathcal{L}^{\otimes k}$ is $\pi-$ample outside of $\pazocal Z$. Moreover, from the short exact sequence:
\[0\to\mathcal{L}(-\lambda)^{\otimes k}\to \mathcal{L}^{\otimes k}\to\mathcal{L}_{\rvert k\pazocal Z}^{\otimes k}\to 0\]
and the vanishing of $R^1\pi_*\mathcal{L}(-\lambda)^{\otimes k}$ (stability), it is enough to show that for any $x\in\pazocal Z$ there exists a section of $\pi_*\mathcal{L}_{\rvert k\pazocal Z}^{\otimes k}$ around $\pi(x)$ which does not vanish in $x$. By definition of $\lambda$, the line bundle $\mathcal{L}\rvert_{\pazocal Z}$ is the pullback along $\psi$ of a line bundle of degree one and non-negative multidegree on $\mathcal T_{\pazocal Z}$; the latter has enough sections.

\noindent\emph{Properties of the fibres}: can be studied after base-change to a generic trait $T$; we may assume that the generic point corresponds to a smooth curve, and the closed point maps to $\pazocal D=\pazocal D_1\cup\pazocal D_2$. Thus
\[\phi_T\colon C_T\to C_T^\prime\]
is a birational contraction satisfying $\phi_*\mathcal O_C=\mathcal O_{C^\prime}$. It follows that $C_T^\prime$ is a normal surface.
In particular, the central fibre is $S1$; it is also generically reduced, being birational to that of $C_T$, thus it is reduced (and Cohen-Macaulay).

Finally, we want to argue that the fibres are Gorenstein outside $\pazocal D_1$. We may assume that the special point of $T$ maps to $\pazocal D_2\setminus\pazocal D_1$. Then $\mathcal L$ is trivial along $Z$, which is therefore contracted to a codimension two locus $Z^\prime$ of $C^\prime_T$. Outside of $Z$, $\phi$ restricts to an isomorphism. The equality of line bundles:
\[\mathcal O_{C^\prime_T}(1)(-\sum p_i)_{|C^\prime_T\setminus Z^\prime}=\omega_{C/T|C_T\setminus Z}=\omega_{C^\prime/T|C^\prime_T\setminus Z^\prime},\]
together with the fact that $\omega_{C^\prime/T}$ is an $S2$ sheaf, shows that that the latter coincides with the line bundle $\mathcal O_{C^\prime_T}(1)(-\sum p_i)$ on the whole of $C^\prime_T$ (Hartogs' Theorem). Thus, $C^\prime_T$ is Gorenstein over $T$.

\noindent\emph{Compatible contraction of $\mathcal T$}: can be performed using the line bundle \[\omega_\mathcal T(D_B)(\lambda_T)(\psi(\mathbf p)),\]
whose pullback to $\widetilde{\mathcal C}$ is $\mathcal L$.

\noindent\emph{Well-posedness}: the construction of $\mathcal{C}^\prime$ and $\mathcal{T}^\prime$ is independent of the choice of markings respecting the weight function by birational rigidity \cite[Lemma 1.15]{Debarre}.
\end{proof}


\subsection{Second step: the pushout}
With a view towards the moduli space of maps, $\pazocal D_1^2$ corresponds to the hyperelliptic component $\pazocal D^{\rm{hypell},k}$, see Section \ref{sec:irrcompos}. Over this locus it is therefore natural to require factorisation through the hyperelliptic cover: in terms of Gorenstein singularities, this produces ribbons. They can also be used to interpolate between isolated singularities over $\pazocal D_1^0$ - this is completely natural from the point of view of piecewise-linear functions on the tropical side, and serves as a correction of the failure of $\mathcal{C}^\prime$ at being Gorenstein. We introduce them over $\pazocal D_1^1$ as well, even though this locus is not in the image of the moduli space of maps: the \emph{bottom line} is that a line bundle of degree one or two on a curve of genus two has enough sections to give a morphism to projective space if and only if it is the dualising line bundle. We also deal with the (non-geometric) case that the weight splits onto two components of weight one and genus one: in this case the canonical has basepoints; we replace the hyperelliptic cover with the restriction of $\psi$ to the relevant subcurve.


Recall that $\pazocal Z'$ was defined as the image of $\pazocal Z$ under $\phi$. First, we prove that the definition of $\pazocal Z'$ commutes with base-change to a generic trait; then, we will show that the pushout construction commutes with such a base-change, and thus we may reduce to the case of surfaces in order to study the singularities of the fibres.

\begin{prop}\label{prop:basechangeZ'}
Let $\pazocal Z'$ be the subscheme of $\mathcal{C}^\prime$ defined by the ideal sheaf $\phi_*\mathcal{O}_{\mathcal C}(-\lambda)$ and supported on $\pazocal D.$
Then we have:
\begin{enumerate}
    \item  $R^1\phi_*\mathcal{O}_{\mathcal C}(-\lambda)=0$, in particular $\phi_*\mathcal{O}_{\mathcal C}(-\lambda)=\operatorname{Fitt}(\phi_*\mathcal{O}_{\pazocal Z})$;
    \item For every generic trait $\Delta\xrightarrow{\iota}\widetilde{\pazocal A}$, the definition of  $\pazocal Z'$ commutes with base change, i.e., $\iota^*\phi_*\mathcal{O}_{\mathcal C}(-\lambda)=\phi_{\Delta *}\mathcal{O}_{\mathcal C_{\Delta}}(-\lambda)$.   
\end{enumerate}
The analogous statements about $\mathcal T^\prime_{\pazocal Z}$ hold as well.
\end{prop}
\begin{proof} The discussion has been somewhat inspired by \cite[\S 1]{Teissier}.
\begin{enumerate}[leftmargin=0cm,itemindent=0.8cm,labelwidth=\itemindent,labelsep=0cm,align=left]
    \item Let $C_s$ be a fiber on which $\phi_s$ is not an isomorphism; in particular $\lambda_s\neq 0$. 
    Working locally on the base, we can choose smooth and disjoint sections $p_1\ldots,p_d$ of $\mathcal C$ respecting the weight function.
    It is enough to prove that $R^1\phi_*\mathcal{O}_{\mathcal C}(-\lambda)(k\mathbf{p})=0$ for $k\gg0.$ Indeed, once we know the latter vanishing, since $\phi$ is an isomorphism around $\mathbf{p},$ we have that $\mathcal{O}_{\mathcal C}(k\mathbf{p})=\phi^*\mathcal{O}_{\mathcal C'}(k\mathbf{p}')$ and thus
     \begin{align*}
     0&=R^1\phi_*\mathcal{O}_{\mathcal C}(-\lambda)(k\mathbf{p})=H^1(R\phi_*\mathcal{O}_{\mathcal C}(-\lambda)(k\mathbf{p}))\\
     &=H^1(R\phi_*\mathcal{O}_{\mathcal C}(-\lambda)\otimes\phi^*\mathcal{O}_{\mathcal C'}(k\mathbf{p}'))=R^1\phi_*\mathcal{O}_{\mathcal C}(-\lambda)\otimes\mathcal{O}_{\mathcal C'}(k\mathbf{p}')
     \end{align*}
     implying the desired vanishing.
     
    From the spectral sequence computing $R\pi'_*\circ R\phi_*$, the five-terms exact sequence:
    \[0\to R^1\pi'_*\phi_*\mathcal{O}_{\mathcal C}(-\lambda)(k\mathbf{p})\to R^1\pi_*\mathcal{O}_{\mathcal C}(-\lambda)(k\mathbf{p})\to\pi'_*R^1\phi_*\mathcal{O}_{\mathcal C}(-\lambda)(k\mathbf{p})\]
    ends there, because the next term would involve an $\operatorname{R}^2\pi^\prime_*(-)$, that vanishes as the fibre dimension is bounded by one.
    
    Notice that if $ R^1\pi_*\mathcal{O}_{\mathcal C}(-\lambda)(k\mathbf{p})\otimes k(s)=0$ for those $s$ where $\phi_s$ is not an isomorphism,
    so is $\pi'_*R^1\phi_*\mathcal{O}_{\mathcal C}(-\lambda)(k\mathbf{p})\otimes k(s).$ 
    
    On the other hand, $R^1\phi_*\mathcal{O}_{\mathcal C}(-\lambda)(k\mathbf{p})$ 
is a coherent sheaf supported on the closed locus $V'$ in $\mathcal C'$ over 
    which the fiber of $\phi$ has positive dimension. The latter is finite over $\widetilde{\pazocal A}$, and thus $\pi'_*R^1\phi_*\mathcal{O}_{\mathcal C}(-\lambda)(k\mathbf{p})\otimes k(s)=0$ implies $R^1\phi_*\mathcal{O}_{\mathcal C}(-\lambda)(k\mathbf{p})\otimes k(x)=0$ for each $x\in V'_s.$
    
    
    Since $R^1\pi_*\mathcal{O}_{\mathcal C}(-\lambda)(k\mathbf{p})$ satisfies cohomology and base change, the vanishing can be checked after restricting to a fibre. Let $Z_s$ denote the support of $\lambda_s$, $C_i$ the trees rooted at the vertices of $\partial\scalebox{0.9}[1.3]{$\triangle$}$, and $e_i$ the first edges encountered in $\scalebox{0.9}[1.3]{$\triangle$}^\circ$. Taking the normalisation of $C_s$ at the nodes corresponding to the edges $e_i$ we get
\[0\to \mathcal{O}_{C_s}(-\lambda)(k\mathbf{p})\to \mathcal{O}_{Z_s}(-\lambda)\oplus\bigoplus_i \mathcal{O}_{C_i}(-\lambda)(k\mathbf{p})\to\bigoplus_i\mathbb C_{e_i} \to 0.\]
The evaluation map on the nodes $e_i$ is clearly surjective at the level of $H^0$, as the line bundle restricted to $C_i$ is very ample for $k$ big enough, and the desired vanishing follows from that of $H^1(\mathcal{O}_{Z_s}(-\lambda_s))$, which can be argued by the definition of $\lambda$ and Serre duality.


\item We already know that the construction of $\mathcal C'$ commutes with base change, so the following diagram is Cartesian:
\[\begin{tikzcd}
\mathcal C_{\Delta}\ar[r,"\phi_{\Delta}"]\arrow[d,"\iota"]\ar[dr,phantom,"\Box"] & \mathcal C'_{\Delta}\arrow[d,"\iota' "]\\
\mathcal C\ar[r,"\phi"] & \mathcal C'.
\end{tikzcd}\]
Furthermore, since the source and target are smooth, $\Delta\hookrightarrow\widetilde{\pazocal A}$ is an l.c.i morphism, and so are $\iota$ and $\iota^\prime$; it thus follows from \cite[Corollary~2.27]{kuznetsov2006hyperplane} that
\[L\iota'^*R\phi_*\mathcal{O}_{\mathcal C}(-\lambda)=R\phi_{\Delta *}L\iota^*\mathcal{O}_{\mathcal C}(-\lambda).\]
On the other hand, the higher pushforward vanishes by the previous point, i.e. $R\phi_*\mathcal{O}_{\mathcal C}(-\lambda)=\phi_*\mathcal{O}_{\mathcal C}(-\lambda)$, and  $L\iota^*\mathcal{O}_{\mathcal C}(-\lambda)=\mathcal{O}_{\mathcal C_{\Delta}}(-\lambda)$ because it is a line bundle, hence the derived statement is equivalent to what we want.

\end{enumerate}
\end{proof}


\begin{prop}\label{prop:Goreinstein+flatness.}
The restriction of $\pazocal Z^\prime$ to $\pazocal D_1$ is a flat family of Gorenstein curves of genus two. Similarly, $\mathcal T^\prime_{\pazocal Z}$ is a flat family of Gorenstein curves of genus zero (i.e. at worst nodal).
\end{prop}

Notice that $\mathcal{C}^\prime$ is not always Gorenstein over $\pazocal D_1$, in particular it can be the decomposable union of $\pazocal Z^\prime$ with some lines. For the proof we need the following:

\begin{lem}\label{lem:Dreduced}
$\pazocal D_1$ is a reduced divisor.
\end{lem}
\begin{proof}
$\pazocal D_1$ is a Cartier divisor in a smooth ambient space, so it is enough to check that it is generically reduced.

The generic point of $\pazocal D_1$ looks like in Figure \ref{fig:genericD1}.
\begin{figure}[h]
 \tikz{
 \draw[fill=black] (-1,0) circle (2pt) --node[right]{\color{blue}{1}} (.5,1.5) (0,0) circle (2pt) --node[right]{\color{blue}{1}} (.5,1.5) (1,0) node{\ldots}  (2,0) circle (2pt) --node[right]{\color{blue}{1}} (.5,1.5);
 \draw[fill=white] (.5,1.5) circle (3pt) node[right]{g=2};
 \draw[red,fill=red] (.5,1.5) circle (1pt);
 }
 \caption{$\lambda$ at the generic point of $\pazocal D_1$.}
 \label{fig:genericD1}
\end{figure}

Therefore, a generic trait with uniformiser $t$ will intersect $\pazocal D_1$ in $(t)$.
 \end{proof}


\begin{proof}[Proof of Proposition~\ref{prop:Goreinstein+flatness.}]

We may change the base to a generic trait $\Delta$ with closed point mapping to the given point of $\pazocal D_1$:
\[\begin{tikzcd}
 C\ar[rr,"\phi"] \arrow[rd,"\pi"] & & C'\ar[dl,"\pi' "]\\
 &\Delta
\end{tikzcd}\]
so $C$ is a smooth surface, $\phi$ is a birational contraction, which is an isomorphism outside the divisor $Z$ defined by $\mathcal I_{Z}=\mathcal{O}_{C}(-\lambda)$.
Notice that  there is a subcurve of $Z$ on which the line bundle defining the contraction is ample, therefore $Z^\prime\subseteq C^\prime$ has pure codimension one. We want to show that $\omega_{Z'}$
is a line bundle and that $\chi(\omega_{Z'})=1$ (equivalently, $p_a(Z')=2$).

Recall that, by Grothendieck duality:
\[\omega_{Z'}=\mathcal E xt^1_{C'}(\mathcal O_{Z'},\omega_{C'})=\mathcal Hom_{C}(\mathcal I_{Z'},\omega_{C'})\rvert_{Z'}.\]

By adjunction for the Cartier divisor $Z\subseteq C$, there is a short exact sequence:
\begin{equation}\label{eqn:adjunctionZ}
0\to\omega_C\to\omega_C(\lambda)\to\omega_Z\to0,
\end{equation}
which stays exact after pushforward along $\phi$ by Grauert-Riemenschneider vanishing \cite[Corollary 2.68]{kollar2008birational}. By applying $\mathcal Hom_{\mathcal O_{C'}}(-,\omega_{C'})$ to the exact sequence:
\[0\to \phi_*\mathcal O_C(-\lambda)\to\mathcal O_{C'}\to\mathcal O_{Z'}\to 0,\]
we obtain the bottom row of the following diagram:
\begin{center}\begin{tikzcd}
0\ar[r] & \phi_*\omega_C\ar[r]\ar[d]&\phi_*\omega_C(\lambda)\ar[r]\ar[d] &\phi_*\omega_Z\ar[d]\ar[r] & 0\\
0\ar[r] & \omega_{C'}\ar[r] & \mathcal Hom_{C}(\phi_*\mathcal O_C(-\lambda),\omega_{C'})\ar[r] &\omega_{Z'}\ar[r] & 0
\end{tikzcd}\end{center}
By Grothendieck duality ($\phi_*\dashv\phi^!$ and $\phi^!\omega_{C^\prime}=\omega_C$) and the snake lemma, we conclude that the vertical arrows are isomorphisms. This implies that $\omega_{Z'}$ is a line bundle if $\phi_*\omega_C(\lambda)$ is.
Since the sections $\mathbf{p}$ are away from $\operatorname{Exc}(\phi)$, by construction \[\phi_*\omega_C(\lambda)=\mathcal{O}_{C'}(1)(-\phi(\mathbf{p}))\]
is a line bundle. We have thus proved that $Z'$ is Gorenstein.

Moreover, $\phi_*\omega_Z=\omega_{Z'}$. Now, to prove that $\chi(\omega_{Z'})=1$ it is enough to prove that $\chi(\omega_{Z'})=\chi(\omega_Z)$. Indeed, over $\pazocal D_1$, the definition of $\lambda$ and adjunction \eqref{eqn:adjunctionZ} show that $\omega_C(\lambda)$ restricts to a line bundle of degree two on $Z$, which is therefore a curve of genus two. Since $\phi_*\mathcal O_Z=\mathcal O_{Z'}$, it is enough to prove the vanishing of $\operatorname{R}^1\phi_*\mathcal O_Z$. By Proposition \ref{prop:basechangeZ'}, $\operatorname{R}^1\phi_*\mathcal O_Z\simeq\operatorname{R}^1\phi_*\mathcal O_C$, therefore the desired statement is equivalent to the fact the $C'$ has rational singularities. This follows from $\phi_*\omega_C=\omega_{C'}$, see for example \cite{kovacs}. We have thus proved that $Z'$ has genus two.

In order to prove the flatness of $\pazocal Z'_{\pazocal D_1}\to\pazocal D_1$, it is sufficient to show that \[\pi'_*(\mathcal{O}_{\pazocal Z'_{\pazocal D_1}}\otimes\mathcal{O}_{\mathcal C'}(n))\]  is a vector bundle on $\pazocal D_1$ for $n$ large enough. Since $\pazocal D_1$ is a reduced divisor (Lemma \ref{lem:Dreduced}), we only have to show that its rank is constant along $\pazocal D_1$. It is easy to see that:
\[\mathcal{O}_{\mathcal C'}(1)_{\rvert\pazocal Z'_{\pazocal D_1^i}}\cong\omega^{i+1}_{\pazocal Z'_{\pazocal D_1^i}},\qquad i=0,1,2.\]

Since $\pazocal Z^\prime_{\pazocal D_1}$ is a curve of genus two, it follows from Riemann-Roch that
\[H^0(Z'_s,\omega_{Z'_s}^{\otimes n(i+1)})=2(i+1)n-1,\]
so it is enough to show that $\mathcal{O}_{\mathcal C'}(1)_{\rvert\pazocal Z'_{\pazocal D_1^i}}$ satisfies cohomology and base-change. 
For this, we observe the short exact sequence:
\begin{center}\begin{tikzcd}
0\ar[r] &\pi'_*(\phi_*\mathcal{O}_{\mathcal C}(-\lambda)\otimes\mathcal{O}_{\mathcal C'}(n))\ar[d,"\cong"]\ar[r] &\pi'_*\mathcal{O}_{\mathcal C'}(n)\ar[r] &\pi'_*\mathcal{O}_{\mathcal C'}(n)\rvert_{\pazocal Z'}\ar[r] & 0\\
& \pi_*(\omega_{\pi}^{\otimes n}(n\mathbf{p})\otimes\mathcal{O}_{\mathcal C}((n-1)\lambda))
\end{tikzcd}\end{center}
As $\mathcal C$ and $\mathcal C^\prime$ are flat over the base, and both $\omega_\pi(\mathbf{p})$ and $\mathcal O_{\mathcal C^\prime}(1)$ are relatively ample, the first two bundles satisfy cohomology and base-change. It follows from a diagram-chase that so does the third. We have thus proved that $\pazocal Z^\prime$ is flat on $\pazocal D_1$.

The statement about $\mathcal T^\prime_{\pazocal Z}$ can be proven in an analogous (but easier) fashion.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm:pushout}]
Noticing that $\pazocal{Z}^\prime\to\mathcal T^\prime_{\pazocal Z}$ is finite, the existence of the pushout as a scheme over $\widetilde{\mathcal A}_2$ follows from results of D. Ferrand \cite{Ferrand}.
We have already proved that the construction of $\mathcal{C}^\prime,\ \pazocal{Z}^\prime,$ and $\mathcal T^\prime_Z$ commutes with pullback to a generic trait. The pushout does as well in virtue of \cite[\href{https://stacks.math.columbia.edu/tag/0ECK}{Tag 0ECK}]{stacks-project}. So, in order to prove that the fibres of $\overline{\mathcal C}$ are Gorenstein, we may work with fibred surfaces, in which case we may apply some results of M. Reid \cite{reid1994nonnormal}.

Following \cite[\S~2.1]{reid1994nonnormal}, $\bar\phi\colon C^\prime\to\overline{C}$ is the normalisation, with conductor:
\[\operatorname{Ann}(\bar\phi_*\mathcal O_{C'}/\mathcal O_{\overline C})=\operatorname{Ann}(\psi_*\mathcal O_{Z'}/\mathcal O_{T'_Z}).\]

Since $\psi$ is a double cover, it is in particular flat, and $\psi_*\mathcal O_{Z'}/\mathcal O_{T'_Z}$ is a line bundle; it follows from \cite[Proposition~2.2]{reid1994nonnormal} that $\overline{C}$ is $S2$.

Moreover, we have seen in the proof of Proposition \ref{prop:Goreinstein+flatness.} that \[\omega_{C'}(Z')=\mathcal Hom_{\mathcal C'}(\mathcal I_{\pazocal Z'},\omega_{\mathcal C'})\] is a line bundle on $C'$. Since $\psi\colon Z'\to T'_Z$ is a double cover of a (nodal chain of) rational curve(s), it follows that the kernel of the canonical map $\psi_*\omega_{Z'}\to\omega_{T'_Z}$ is a line bundle as well. The criterion of \cite[Corollary~2.8~(iv)]{reid1994nonnormal} allows us to conclude that $\overline C$ is Gorenstein.

Finally, the statement about weights is obvious from the construction of $\lambda$.
\end{proof}


\subsubsection{Markings}\label{sec:admmarkings}
To avoid  overloading the notation and the exposition,  we have so far considered only weighted admissible covers without markings. However, with the application to stable maps in mind, markings are necessary to impose cohomological constraints using the evaluation maps.
Our construction extends to the marked version essentially unchanged. We can  consider:
\begin{dfn}
  A weighted admissible cover with markings consists of: \[\left(\psi\colon(C,D_R,\mathbf{x})\to(T,D_B,\mathbf{y}=\psi(\mathbf{x})),w\colon V(\scalebox{0.8}[1.3]{$\sqsubset$})\to\mathbb N\right)\]  such that $D_R$ and $\mathbf{x}$ are separately disjoint (multi-)sections of $C$, and $T$ is weighted-stable, i.e every weight-zero component has at least three special points between markings and nodes. We denote the moduli space of weighted admissible covers with $n$ markings by $\mathcal A_{2,n}^{\text{wt}}$.
\end{dfn}
Markings are represented by infinitely long legs on $\scalebox{0.8}[1.3]{$\sqsubset$}$. They play no role in the alignment: admissible functions will have slope $1$ along them, and the infinite legs will be subdivided accordingly. In particular, if a marking is supported on a vertex of $\scalebox{0.9}[1.3]{$\triangle$}^\circ$, then we may \emph{sprout} (blow-up) the marking as many times as it is necessary for its strict transform to be supported on a vertex of $\partial\scalebox{0.9}[1.3]{$\triangle$}$.
\begin{thm}
There exists a logarithmically \'etale modification $\widetilde {\mathcal A}_{2,n}\to\mathcal A_{2,n}^{\text{wt}}$ parametrising aligned weighted admissible covers with markings. The moduli space $\widetilde {\mathcal A}_{2,n}$ is a logarithmically smooth DM stack with locally free logarithmic structure.
\end{thm}

In particular, the strict transform of the markings never touches the singularity.
\begin{thm}
 There is a diagram of flat families of projective, Gorenstein curves of arithmetic genus two, with $n$ corresponding disjoint sections of the smooth locus:
 \begin{center}\begin{tikzcd}
 & \widetilde{\mathcal C}\ar[dr]\ar[dd]\ar[dl] & \\
 \mathcal C\ar[dr] & & \overline{\mathcal C}\ar[dl] \\
 & \widetilde{\mathcal A}_{2,n}\ar[uu,bend right,"\tilde{\mathbf x}"] &
 \end{tikzcd}\end{center}
 Moreover, the weight of the minimal subcurve of genus two in $\overline{\mathcal C}$ is at least three.
\end{thm}

\section{A modular desingularisation of $\M{2}{n}{\mathbb P^r}{d}^\text{main}$}\label{sec:maps}

\subsection{Irreducible components}\label{sec:irrcompos}
We draw the weighted dual graph of the general member of all possible irreducible components of $\overline{\pazocal{M}}_2(\mathbb P^r,d)$. \begin{comment}If there were any markings, they could distribute freely among the vertices, so that irreducible components would be labelled by partitions not only of the degree, but of the markings as well.\end{comment}
Our running convention is that a white vertex corresponds to a contracted component, a gray one to a genus two subcurve covering a line two-to-one, and a black vertex corresponds to a non-special subcurve. Vertices are labelled with their genus and weight.
\begin{enumerate}[leftmargin=.6cm]
 \item \emph{main} is the closure of the locus of maps from a smooth curve of genus two;
 \item $\pazocal D^{(d_1,\ldots,d_k)}=\left\{\Dk\right\}$
 \item ${}^{\rm{hyp}}\!\pazocal D^{(d_1,\ldots,d_k)}=\left\{\hypell\right\}$
 \item ${}^{d_0}\!\pazocal E^{(d_1,\ldots,d_k)}=\left\{\Ek\right\}$
 \item ${}^{(d_{1,1},\ldots,d_{1,k_1})}\!\pazocal E^{d_0}\pazocal E^{(d_{2,1},\ldots,d_{2,k_2})}=\left\{\Ekk\right\}$
 \item ${}^{\rm{br}=d_0}\!\pazocal E^{(d_1,\ldots,d_k)}=\left\{\pacman\right\}$
\end{enumerate}
This is taken from the first author's PhD thesis \cite{Battistella-thesis}, and is implicit in \cite{HLN}.

\begin{comment}
\subsection{Low degree}\marginpar{{\textcolor{red}{the section below is wrong!}}}
\begin{lem}
 The main component of $\overline{\pazocal{M}}_2(\mathbb P^1,2)$ is isomorphic to $\mathcal A_2$.
\end{lem}
\begin{proof}
 There is a morphism
 \[\sigma\colon \mathcal A_2\to\overline{\pazocal{M}}_2(\mathbb P^1,2)\]
 that stabilises the source curve (and the target correspondingly):
 \begin{center}\begin{tikzcd}
 C\ar[d,"f"]\ar[r,"st"] & C^s\ar[d,"\phi_{\lvert\omega\rvert}"] \\
 T\ar[r,"st"] & \mathbb P^1.
 \end{tikzcd}\end{center}
 $\sigma$ factors through the main component. Moreover, the composition
 \[\mathcal A_2\xrightarrow{\sigma}\overline{\pazocal{M}}_2(\mathbb P^1,2)\xrightarrow{fgt}\overline{\pazocal{M}}_2\]
 is an isomorphism. It follows that $\sigma$ is as well (by Zariski's Main Theorem).
\end{proof}
\begin{cor}
 The main component of $\overline{\pazocal{M}}_2(\mathbb P^r,2)$ is a fibration
 \begin{center}\begin{tikzcd}
 \mathcal A_2\ar[r] & \overline{\pazocal{M}}_2^{\text{main}}(\mathbb P^r,2)\ar[d] \\
 {} & \operatorname{Gr}(2,r+1).
 \end{tikzcd}\end{center}
\end{cor}
\end{comment}

\subsection{Factorisation through a Gorenstein curve}
\begin{dfn}
 Let $\mathcal A_{2,n}(\mathbb P^r,d)$ be defined by the following Cartesian diagram:
 \begin{center}\begin{tikzcd}
 \mathcal A_{2,n}(\mathbb P^r,d)\ar[d]\ar[r]\ar[dr,phantom,"\Box"] & \overline{\pazocal M}_{2,n}(\mathbb P^r,d)\ar[d]\\
 \mathcal A_{2,n}^{\text{wt}}\ar[r] & \mathfrak{M}_{2,n}^{\text{wt}}.
 \end{tikzcd}\end{center}
 
 \noindent It can be described as a space of maps and admissible covers with the same source:
 \begin{center}\begin{tikzcd}
 (C,D_R,\mathbf{x})\ar[d,"\psi"]\ar[r,"f"] & \mathbb P^r\\
 (T,D_B,\psi(\mathbf{x}))
 \end{tikzcd}\end{center}
 subject to the stability condition that, if $\sigma\colon C\to C$ is the hyperelliptic involution,
 \[\omega_C(\mathbf{x})\otimes f^*\mathcal O(2)\otimes\sigma^*f^*\mathcal O(2)\]
 is relatively ample. 
\end{dfn}
\begin{rem}\label{rem:mainbirational}
 On the locus of maps from a smooth curve, $\mathcal A_{2,n}(\mathbb P^r,d)\to\overline{\pazocal M}_{2,n}(\mathbb P^r,d)$ is an isomorphism; therefore, their main components are birational.
\end{rem}
\begin{dfn}
 More generally, for a polarised variety $(X,\mathcal O_X(1))$ and $\beta\in H^+_2(X,\mathbb Z)$, we can define the moduli space $\mathcal A_{2,n}(X,\beta)$ of admissible maps to $X$.
\end{dfn}

\begin{dfn}
 Let $\widetilde{\mathcal A}_{2,n}(X,\beta)$ denote the fibre product:
 \begin{center}\begin{tikzcd}
 \widetilde{\mathcal A}_{2,n}(X,\beta)\ar[d]\ar[r]\ar[dr,phantom,"\Box"] & \mathcal A_{2,n}(X,\beta)\ar[d]\\
 \widetilde{\mathcal A}_{2,n}\ar[r] & \mathcal A_{2,n}^{\text{wt}}.
 \end{tikzcd}\end{center}
 We call it the moduli space of aligned admissible maps.
\end{dfn}
\begin{rem}\label{rem:logetale}
 $\widetilde{\mathcal A}_{2,n}(X,\beta)$ is logarithmically \'{e}tale over $\mathcal A_{2,n}(X,\beta)$. It comes with universal structures:
 \begin{center}\begin{tikzcd}[cramped]
  & \widetilde{\mathcal C}\ar[dl,"p"]\ar[dr]\ar[drr,"\tilde f"] & & \\
  \overline{\mathcal C} & & \mathcal C\ar[r,"f" below]\ar[d,"\psi"] & X \\
  & & \mathcal T &
 \end{tikzcd}\end{center}
\end{rem}
\begin{dfn}
 Let $\pazocal{V\!Z}_{2,n}(X,\beta)\subseteq\widetilde{\mathcal A}_{2,n}(X,\beta)$ be the locus of maps such that $\tilde f\colon\widetilde{\mathcal C}\to X$ factors through a map $\bar f\colon\overline{\mathcal C}\to X$. We call it the moduli space of aligned admissible maps satisfying the \emph{factorisation property}\footnote{The notation is reminiscent of the celebrated desingularisation of $\M{1}{n}{\mathbb P^r}{d}^{\text{main}}$ due to R.Vakil and A. Zinger \cite{VZ}.}.
\end{dfn}
\begin{thm}
 $\pazocal{V\!Z}_{2,n}(X,\beta)\subseteq\widetilde{\mathcal A}_{2,n}(X,\beta)$ is a closed substack. If $X$ is smooth, there is a perfect obstruction theory:
 \[(\operatorname{R}^\bullet\bar\pi_*\bar f^*T_X)^\vee\to\mathbb L^\bullet_{\pazocal{V\!Z}_{2,n}(X,\beta)/\widetilde{\mathcal A}_{2,n}}\]
 endowing $\pazocal{V\!Z}_{2,n}(X,\beta)$ with a virtual fundamental class in $A_{\operatorname{vdim}}(\pazocal{V\!Z}_{2,n}(X,\beta))$, where:
 \[\operatorname{vdim}(\pazocal{V\!Z}_{2,n}(X,\beta))=3-\dim(X)+n-K_X\cdot\beta.\]
\end{thm}
\begin{proof}
 For the first claim, we refer the reader to \cite[Theorem 4.3]{RSPW1}.
 The second claim goes back to K. Behrend and B. Fantechi \cite[Proposition 6.3]{BF}.
\end{proof}


\begin{dfn}
 Let $\operatorname{ev}_i\colon \pazocal{V\!Z}_{2,n}(X,\beta)\to X$ denote the evaluation map at the $i$-th marked point, $i=1,\ldots,n$. Let $\alpha_1,\ldots,\alpha_n\in H^*(X)$ be cohomology classes on the target manifold. The \emph{reduced genus two Gromov-Witten invariants} are defined as:
 \[\langle\alpha_1,\ldots,\alpha_n\rangle^{X,\text{red}}_{2,\beta,n}=\int_{[\pazocal{V\!Z}_{2,n}(X,\beta)]^\text{vir}}\operatorname{ev}_1^*\alpha_1\cup\ldots\cup\operatorname{ev}_n^*\alpha_n.\]
 
\end{dfn}
\begin{thm}
 For $d\geq 3$, $\pazocal{V\!Z}_{2,n}(\mathbb P^r,d)$ is a desingularisation of $\overline{\pazocal M}_{2,n}(\mathbb P^r,d)^\text{main}$.
\end{thm}
\begin{proof}
 Consider the factorisation:
 \[ \pazocal{V\!Z}_{2,n}(\mathbb P^r,d)\to \mathfrak{P}ic_{\widetilde{\mathcal A}_{2,n}}\to\widetilde{\mathcal A}_{2,n}. \]
 Obstructions to the first map can be found in $H^1(\overline{C},L)$ - where $L=\bar f^*\mathcal O_{\mathbb P^r}(1)$ -, which vanishes for degree reasons (see the final claim of Theorem \ref{thm:pushout}). Obstructions to the second map lie in $H^2(\overline{C},\mathcal O)$, which vanishes by dimension reasons. The map is therefore unobstructed. The base is smooth by Theorem \ref{thm:tildeA}. We conclude that $\pazocal{V\!Z}_{2,n}(\mathbb P^r,d)$ is smooth as well. Since it is proper and it contains the locus of maps from a smooth curve as an open dense, $\pazocal{V\!Z}_{2,n}(\mathbb P^r,d)\to \overline{\pazocal M}_{2,n}(\mathbb P^r,d)^\text{main}$ is birational (see Remarks \ref{rem:mainbirational} and \ref{rem:logetale}).
\end{proof}


\begin{rem}
 A posteriori, we note that aligning and the factorisation property do not alter the main component of $\overline{\pazocal M}_2(\mathbb P^r,2)$, which therefore is already smooth.
\end{rem}



