\section{Introduction}

It is an important problem in algebraic geometry and in differential geometry to decide if a given Fano variety $X$ admits a K\"ahler--Einstein (KE) metric. The Yau--Tian--Donaldson (YTD) Conjecture predicts that the existence of a KE metric on $X$ is equivalent to the K-polystability of $X$.
Using Cheeger--Colding--Tian theory, the YTD Conjecture was first proved when $X$ is smooth \cite{chen_donaldson_sun, tian, berman_polystability}, when $X$ is $\mathbb{Q}$-Gorenstein smoothable \cite{LWX19, SSY16}, or when $X$ has dimension $2$ \cite{ltw_1}. Later, a different method, namely the variational approach, was introduced in \cite{BBJ15}. The analytic side of the variational approach was completed in \cite{ltw_2, Li19} which shows that a $\mathbb{Q}$-Fano variety $X$, that is, a Fano variety with klt singularities, admits a KE metric if and only if $X$ is reduced uniformly K-stable, a concept introduced in \cite{His16} as an equivariant version of uniform K-stability (see also \cite{XZ19}). Recently, using purely algebro-geometric methods, the work \cite{LXZ21} establishes the equivalence between K-polystability and reduced uniform K-stability. This work, combining with the variational approach, proves the YTD Conjecture for all $\mathbb{Q}$-Fano varieties.



K-stability of del Pezzo surfaces which are quasi-smooth hypersurfaces in weighted projective $3$-spaces has been studied extensively.
Johnson and Koll\'ar \cite{kollar_johnson} classified those which are anticanonically polarised (i.e.\ have Fano index $1$) and decided the existence of a KE metric on many of these, by using Tian's criterion which relates KE metrics to global log canonical thresholds (also called $\alpha$-invariants) \cite{tian_alpha, nadel_annals, demailly_kollar, cheltsov_lct, odaka_sano_alpha, Fuj19b}.
This method was applied to most of these del Pezzo surfaces by Araujo~\cite{araujo_del_pezzo}, Boyer--Galicki--Nakamaye~\cite{boyer_galicki_nakamaye}, and Cheltsov--Park--Shramov~\cite{cheltsov_park_shramov_exceptional}.
One case was missing and was finally solved in \cite{cheltsov_park_shramov_delta} by using delta invariants
(see \cite{fujita_odaka_delta, blum_jonsson}).

The (non-)existence of KE metrics on many del Pezzo surfaces which are quasi-smooth hypersurfaces in weighted projective $3$-spaces with Fano index $\geq 2$ has been studied in \cite{cheltsov_park_shramov_exceptional, cheltsov_park_shramov_delta, del_pezzo_zoo, kim_won}.

In this paper, we study K-polystability of quasi-smooth degree $2a$ hypersurfaces in the weighted projective space $\PP(1,1,a,a)$. When $a\in \{2,4\}$, such del Pezzo surfaces are $\mathbb{Q}$-Gorenstein smoothable, and their K-polystability was determined by Mabuchi--Mukai \cite{MM93} and Odaka--Spotti--Sun \cite{odaka_spotti_sun} (see Remark \ref{rem:a=2or4}). To the authors' knowledge it is not known if they are K-polystable for an integer $a=3$ or $a \geq 5$.
In \cite{kim_won} Kim and Won conjecture that these surfaces are K-polystable and not K-stable.

Our main result relates the K-polystability (resp.\ K-semistability) of degree $2a$ hypersurfaces in $\PP(1,1,a,a)$ to GIT polystability (resp.\ GIT semistability) of degree $2a$ binary forms (see \cite[Chapter~4]{git}).

\begin{theorem} \label{thm:k=git}
	Let $a \geq 2$ be an integer and 
	let $\bP(1,1,a,a)$ be the weighted projective space with coordinates $[x, y, z, w]$ with weights $\deg x = \deg y = 1$ and $\deg z = \deg w = a$. Let $X$ be a hypersurface of degree $2a$ in $\bP(1,1,a,a)$.
	
	Then $X$ is K-semistable (resp.\ K-polystable) if and only if, after an automorphism of $\bP(1,1,a,a)$, the equation of $X$ is given by $z^2+w^2+g(x,y)=0$ where $g\neq 0$ is GIT semistable (resp.\ GIT polystable) as a degree $2a$ binary form.
	Moreover, $X$ is not K-stable.
\end{theorem}

As a consequence we prove the K-polystability of quasi-smooth hypersurfaces in $\PP(1,1,a,a)$ of degree $2a$, hence partially confirming \cite[Conjecture~1.3]{kim_won}.

\begin{cor} \label{cor:k-poly}
Let $a \geq 2$ be an integer and let $X$ be a degree $2a$ quasi-smooth hypersurface in $\PP(1,1,a,a)$. Then $X$ is K-polystable and not K-stable. Moreover, $X$ admits a KE metric.
\end{cor}

Recently the result of this corollary has been independently  announced by Viswanathan using different methods.

It is possible to give a proof of K-polystability for a general hypersurface in $\PP(1,1,a,a)$ of degree $2a$, when $a$ is odd, by analysing the deformation theory of the toric surface appearing in Proposition~\ref{prop:toric} similarly to \cite{ask_petracci} and without using Theorem~\ref{thm:k=git}.




\subsection*{Notation and conventions}
We always work over $\CC$.
A \emph{del Pezzo surface} is a normal projective surface whose anticanonical divisor is $\QQ$-Cartier and ample.
Every toric variety we consider is normal.
We do not even try to write down the definitions of K-(poly/semi)stability of Fano varieties and of log Fano pairs:
we refer the reader to the excellent survey \cite{xu_survey}, the paper \cite{ADL19}, and to the references therein.


\subsection*{Acknowledgements}
The second author wishes to thank Anne-Sophie Kaloghiros for many fruitful conversations and Yuji Odaka for helpful e-mail exchanges;
he is grateful also to Ivan Cheltsov and Jihun Park for useful remarks on an earlier draft of this manuscript and for sharing a preliminary version of \cite{kim_won}. The first author is partially supported by the NSF Grant DMS-2001317.




\section{Proofs}

In what follows $a$ is a fixed integer greater than $1$.
We consider the weighted projective space $\PP(1,1,a,a)$ with coordinates $[x, y, z, w]$ with weights $\deg x = \deg y = 1$ and $\deg z = \deg w = a$.


\begin{comment}
\subsection{Some surface singularities}

If $a$ and $b$ are integers such that $1 \leq b < a$ and $\gcd(a,b)=1$, then $\frac{1}{a}(1,b)$ denotes the isolated surface singularity given by the quotient of $\AA^2$ under the action of $\boldsymbol{\mu}_a$ with weights $(1,b)$. It is a toric affine variety.

\subsubsection{} Fix an integer $a \geq 2$.
The singularity $\frac{1}{a}(1,a-1)$ is the hypersurface 
\[
U = \Spec \CC[x,y,z] / (xy-z^a).
\]
It is Gorenstein and its miniversal ($\QQ$-Gorenstein) deformation is given by the equation 
\[
xy - z^a - \sum_{i=2}^a t_i z^{a-i} = 0
\]
over the smooth germ $\CC \pow{t_2, \dots, t_a}$. 
The sheaf $\cTqG{1}{U}$ is supported on the singular point and has length $a-1$.
 $U$ is a toric variety: consider
the action of the torus $(\mathbb{G}_\mathrm{m})^2$ on $U$ given by
\[
(\lambda_1, \lambda_2) \cdot (x,y,z) = ( \lambda_1 x , \lambda_1^{-1} \lambda_2^a y, \lambda_2 z).
\]
Thanks to this action, the vector space $\cTqG{1}{U}$ is a representation of $(\mathbb{G}_\mathrm{m})^2$
and it splits into the direct sum of the irreducible representations of $(\mathbb{G}_\mathrm{m})^2$ which correspond to the characters
\[
(0,-2), \dots, (0,-a) \in \ZZ^2
\]
of $(\mathbb{G}_\mathrm{m})^2$.

\subsubsection{}
Fix an integer $a \geq 3$ such that $a \neq 4$.
The singularity $\frac{1}{a}(1,1)$ is the cone over the rational normal curve of degree $a$ and is $\QQ$-Gorenstein rigid, so its sheaf $\cTqG{1}{}$ is zero.

\subsubsection{}
The base of miniversal $\QQ$-Gorenstein deformation of the singularity $\frac{1}{4}(1,1)$ is smooth and of dimension $1$.
\end{comment}

\begin{proposition} \label{prop:toric}
If $Y$ is the hypersurface in $\PP(1,1,a,a)$ defined by the equation $zw - x^a y^a  = 0$, then $Y$ is a K-polystable toric del Pezzo surface.	
\end{proposition}

\begin{proof}
	We fix the lattice $N = \ZZ^2$ and its dual $M = \Hom_\ZZ(N,\ZZ)$. Elements of $N$ will be columns and elements of $M$ will be rows. 

	
	Let $Q$ be the convex hull of the points 
	\begin{equation*}
	(0,0), \ (0,1), \ (a^{-1},0), \ ( -a^{-1},1)
	\end{equation*}
     in $M_\RR$.
	Let $\Sigma$ be the inner normal fan of $Q$; thus $\Sigma$ is the complete normal fan in $N$ whose rays are generated by the vectors
	\begin{equation} \label{eq:vertices_of_P}
	\begin{pmatrix}
	a \\ 1
	\end{pmatrix},
	\
	\begin{pmatrix}
	0 \\ 1
	\end{pmatrix},
	\
	\begin{pmatrix}
	-a \\ -1
	\end{pmatrix},
	\
	\begin{pmatrix}
	0 \\ -1
	\end{pmatrix}.
	\end{equation}
	We want to show that $Y$ is the toric variety associated to the fan $\Sigma$.
	
	Provisionally, let $\toric{\Sigma}$ denote the toric variety associated to $\Sigma$.
	Consider the cone $\tau$ in $M \oplus \ZZ$ spanned by $Q \times \{1\}$.
	Consider the finitely generated monoid $\tau \cap (M \oplus \ZZ)$ and the semigroup algebra $\CC[\tau \cap (M \oplus \ZZ)]$, 
	which is $\NN$-graded via the projection $M \oplus \ZZ \onto \ZZ$.
	Toric geometry says that $\toric{\Sigma} = \Proj \CC[\tau \cap (M \oplus \ZZ)]$.
	One can see that the minimal set of generators of the semigroup $\tau \cap (M \oplus \ZZ)$ is made up of the vectors
	\begin{equation*}
	(0,0,1), \ (0,1,1), \ (1,0,a), \ (-1,a,a);
	\end{equation*}
	these vectors satisfy a unique relation:
	\[
	a(0,0,1) + a(0,1,1) = (1,0,a) + (-1,a,a).
	\]
	Hence the $\NN$-graded ring $\CC[\tau \cap (M \oplus \ZZ)]$ coincides with $\CC[x,y,z,w] / (zw - x^a y^a)$, where $\deg x = \deg y = 1$ and $\deg z = \deg w = a$. Therefore $Y = \toric{\Sigma}$.
	
	The vectors in \eqref{eq:vertices_of_P} are the vertices of a polytope $P$ in $N$. This implies that $Y$ is a del Pezzo surface, i.e.\ $-K_Y$ is $\QQ$-Cartier and ample.
	
	Let $P^\circ$ be the polar of $P$; thus $P^\circ$ is the convex hull of $(0, \pm 1)$ and $\pm (\frac{2}{a}, -1)$ in $M_\RR$. 
	The polygon $P^\circ$ is the moment polytope of the toric boundary of $Y$, which is an anticanonical divisor.
	Since $P$ is centrally symmetric, also $P^\circ$ is centrally symmetric, thus the barycentre of $P^\circ$ is the origin. By \cite{berman_polystability} $Y$ is K-polystable.
\end{proof}

	\begin{comment}
	$Q = \frac{1}{2} \left(P^\circ + \left(0, 1 \right)\right)$ with vertices
	

\medskip




\end{comment}




\begin{comment}

\begin{proposition} \label{prop:hypersurfaces_are_general_in_moduli}
Let $a \geq 1$ be an odd integer and let $X$ be a quasi-smooth hypersurface in $\PP(1,1,a,a)$ of degree $2a$.
Then every small $\QQ$-Gorenstein deformation of $X$ is a quasismooth hypersurface in $\PP(1,1,a,a)$ of degree $2a$.
\end{proposition}



\begin{remark}
The conclusion of Proposition~\ref{prop:hypersurfaces_are_general_in_moduli} does not hold for $a=2$ and for $a=4$: indeed, in these two cases a quasismooth hypersurface in $\PP(1,1,a,a)$ of degree $2a$ is singular and $\QQ$-Gorenstein smoothable. We do not know what happens for even integers $a \geq 6$.
\end{remark}

\begin{proof}[Proof of Proposition~\ref{prop:hypersurfaces_are_general_in_moduli}]
	The case $a=1$ is well known and can be easily proved by adapting what follows, without having singularities and without introducing stacks.
	Thus we assume $a \geq 2$.
	
	For brevity let $\PP$ denote the variety $\PP(1,1,a,a)$. This is a normal projective $3$-fold whose singular locus is a smooth rational curve along which $\PP$ has transverse $\frac{1}{a}(1,1)$ singularities.
	Since $\PP$ is a normal variety with quotient singularities, there exists a unique smooth Deligne--Mumford stack $\cP$ with coarse moduli space $\PP$ such that  $\cP \to \PP$ is an isomorphism in codimension $1$ (see  \cite[(2.8)]{vistoli} and \cite[\S4.1]{fmn_toric}).
	Thus the stacky locus of $\cP$ coincides with the singular locus of $\PP$.
	In this particular case, $\cP$ is a quotient stack:
	\[
	\cP = \left[ \AA^4 \setminus \{ 0 \} \ / \ \mathbb{G}_\mathrm{m}   \right]
	\]
	where $\mathbb{G}_\mathrm{m}$ acts linearly with weights $1,1,a,a$.
	
		
	$X$ has $2$ singular points of type $\frac{1}{a}(1,1)$. Let $\mathcal{X}$ be the smooth Deligne--Mumford stack with coarse moduli space $X$ such that $\mathcal{X} \to X$ is an isomorphism in codimension $1$. The stacky locus of $\mathcal{X}$ consists of the $2$ singular points of $X$, where there is a stabiliser $\boldsymbol{\mu}_a$.

	
	We have a cartesian diagram
	\[
	\xymatrix{
	\mathcal{X} \ar[d] \ar[r] & \cP \ar[d] \\
	X \ar[r] & \PP
	}
	\]
	where the horizontal arrows are closed embeddings of effective Cartier divisors and the vertical arrows are the map from a Deligne--Mumford stack to its coarse moduli space.
	
	Since $a$ is odd, the canonical (also called `index one') cover of $\frac{1}{a}(1,1)$ is $\AA^2$. This implies that $\mathcal{X} \to X$ and $\cP \to \PP$ are canonical cover stacks \cite{abramovich_hassett}. Therefore $\QQ$-Gorenstein deformations of $X$ are the deformations which are induced by deformations of $\mathcal{X}$: $\DefqG{X} = \Def{\mathcal{X}}$.
	
	Let $\Hilb{X}{\PP}$ denote the infinitesimal Hilbert functor of the closed embedding $X \into \PP$. We need to show that the natural forgetful map
	\[
	\Hilb{X}{\PP} \longrightarrow \DefqG{X}
	\]
	is smooth.
	This is equivalent to show that
		\[
	\Hilb{\mathcal{X}}{\cP} \longrightarrow \Def{\mathcal{X}}
	\]
	is smooth. As $\mathcal{X}$ and $\cP$ are smooth, from an easy argument in deformation theory (see \cite[Proposition~3.2.9]{sernesi}) it is enough to show that
	$H^1(\cT_\cP \vert_{\mathcal{X}})$ vanishes, where $\cT_\cP$ is the tangent sheaf of $\cP$.
	
The Euler sequence
\[
0 \longrightarrow \cO_\cP \longrightarrow \cO_\cP(1)^{\oplus 2} \oplus \cO_\cP(a)^{\oplus 2} \longrightarrow \cT_\cP \longrightarrow 0
\]
consists of locally free sheaf on $\cP$.
By restricting to $\mathcal{X}$ we get the short exact sequence
\[
0 \longrightarrow \cO_\mathcal{X} \longrightarrow \cO_\mathcal{X}(1)^{\oplus 2} \oplus \cO_\mathcal{X}(a)^{\oplus 2} \longrightarrow \cT_\cP \vert_{\mathcal{X}} \longrightarrow 0.
\]
From the vanishing of $\rH^1(\cO_\mathcal{X}(1))$, of $\rH^1(\cO_\mathcal{X}(a))$, and of $\rH^2(\cO_\mathcal{X})$ we deduce the vanishing of $\rH^1(\cT_\cP \vert_{\mathcal{X}})$.
	\end{proof}
\end{comment}



\begin{remark}
\begin{enumerate}
 \item Another way to show K-polystability of $Y$ is by realising $Y\cong (\bP^1\times\bP^1)/(\mathbb{Z}/a\mathbb{Z})$, where the $\mathbb{Z}/a\mathbb{Z}$-action on $\bP^1\times\bP^1$ is given by 
 \[
  \zeta\cdot ([u_0,u_1],[v_0,v_1]):=([\zeta u_0, u_1],[\zeta^{-1} v_0,v_1])\quad \textrm{with }\zeta=e^{\frac{2\pi i}{a}}.
 \]
 Since the above action is free away from finitely many points, and it preserves the product of Fubini-Study metrics on $\bP^1\times\bP^1$, we know that $Y$ admits a KE metric and hence is K-polystable by \cite{berman_polystability}.

 \item A degree $2a$ hypersurface in $\PP(1,1,a,a)$ is defined by an equation
	\[
	q(z,w) + f(x,y) z + h(x,y) w + g(x,y) = 0
	\]
	where $q$ is a quadratic form, $f$ and $h$ are forms of degree $a$, and $g$ is a form of degree $2a$.
	With an automorphism of $\PP(1,1,a,a)$ which is induced by a linear change of the coordinates $z,w$, we can diagonalise the quadratic form $q$, so that the term $zw$ disappears.
	Furthermore, if $q$ has full rank, with an automorphism of $\PP(1,1,a,a)$ induced by $z \mapsto z + \frac{f}{2}$ and $w \mapsto w + \frac{h}{2}$, the equation becomes
	\[
	z^2 + w^2 + g(x,y) = 0.
	\]
\end{enumerate}
\end{remark}





\begin{proof}[Proof of Theorem~\ref{thm:k=git}]
	We start from the ``if'' part. 
	Suppose $X\subset\bP(1,1,a,a)$ is defined by the equation $z^2+w^2+g(x,y)=0$ with $g\neq 0$. Then the ``if'' part states that $X$ is K-semistable (resp.\ K-polystable) if $g$ is GIT semistable (resp.\ GIT polystable).
	
	By forgetting the $w$-coordinate, we obtain a double cover $\pi: X\to \bP(1,1,a)$ with branch locus $D=(z^2+g(x,y)=0)$. Thus by \cite{LZ20, Zhu20} we know that $X$ is K-semistable (resp.\ K-polystable) if and only if $(\bP(1,1,a),\frac{1}{2}D)$ is K-semistable (resp.\ K-polystable).
	
	Let us assume for the moment that $g$ is an arbitrary degree $2a$ binary form. Denote by $D_0:=(z^2=0)$ as a divisor on $\bP(1,1,a)$.  It is clear that $\bP(1,1,a)$ is the projective cone over $\bP^1$ with polarization $\cO_{\bP^1}(a)$, and $\frac{1}{2}D_0$ is the section at infinity. Since $\bP^1$ is K\"ahler--Einstein,  \cite[Proposition 3.3]{LL19} shows that $(\bP(1,1,a), (1-\frac{r}{2})\frac{1}{2}D_0)$ admits a conical KE metric, where $r\in \mathbb{Q}_{>0}$ satisfies $\cO_{\bP^1}(a)\sim_{\mathbb{Q}} -r^{-1} K_{\bP^1}$, i.e. $r=\frac{2}{a}$. By computation, $(1-\frac{r}{2})\frac{1}{2}=\frac{a-1}{2a}$. Thus $(\bP(1,1,a), \frac{a-1}{2a}D_0)$ admits a conical KE metric and hence is K-polystable.
	It is clear that under the $\bG_m$-action $\sigma$ on $\bP(1,1,a)$ given by $\sigma(t)\cdot [x,y,z]=[x,y,tz]$, the log Fano pair $(\bP(1,1,a), \frac{a-1}{2a}D)$ specially degenerates to $(\bP(1,1,a), \frac{a-1}{2a}D_0)$ as $t\to 0$. Thus by openness of K-semistability \cite{BLX19, Xu19} we know that $(\bP(1,1,a), \frac{a-1}{2a}D)$ is K-semistable. 
	
	Next, we assume that $g\neq 0$ is GIT semistable. By GIT of binary forms, we know that each linear factor in $g(x,y)$ has multiplicity at most $a$. In other words, the curve $D$ has only $A_{k-1}$-singularities (i.e.\ locally analytically given by $x^2+y^k=0$) where $k\leq a$. Thus we have that $\lct(\bP(1,1,a);D)\geq \frac{1}{2}+\frac{1}{a}=\frac{a+2}{2a}$. This implies that $(\bP(1,1,a), \frac{a+2}{2a}D)$ is a log canonical log Calabi--Yau pair. Thus interpolation for K-stability \cite[Proposition 2.13]{ADL19} implies that $(\bP(1,1,a), \frac{1}{2}D)$ is K-semistable. 
	
	Next, we assume that $g\neq 0$ is GIT polystable. There are two cases: $g$ is strictly GIT polystable (i.e.\ GIT polystable but not GIT stable), or $g$ is GIT stable. In the first case, under a suitable coordinate we may write $g(x,y)=x^a y^a$. Thus the double cover $X$ is toric, and as shown in Proposition~\ref{prop:toric} $X$ is K-polystable. In the second case, we know that each linear factor in $g(x,y)$ has multiplicity at most $a-1$. Thus the curve $D$ has only $A_{k-1}$-singularities where $k\leq a-1$. Thus we have that $\lct(\bP(1,1,a);D)\geq \frac{1}{2}+\frac{1}{a-1}>\frac{a+2}{2a}$, which implies that $(\bP(1,1,a), \frac{a+2}{2a}D)$ is a klt log Calabi--Yau pair. Thus interpolation for K-stability \cite[Proposition 2.13]{ADL19} implies that $(\bP(1,1,a), \frac{1}{2}D)$ is K-stable. This finishes the proof of the ``if'' part.
	
	Next, we treat the ``only if'' part. In fact, this follows from moduli comparison arguments as in \cite{ADL19}. Let $\mathbf{A}:=H^0(\bP^1, \cO_{\bP^1}(2a))$ be the affine space parametrizing degree $2a$ binary forms.
	Let $\mathbf{A}^{\rm ss}\subset \mathbf{A}\setminus\{0\}$ be the open subset of GIT semistable binary forms.
	Consider the universal family of weighted hypersurfaces $\mathcal{X}\to \mathbf{A}^{\rm ss}$ where $\mathcal{X}\subset \bP(1,1,a,a)\times \mathbf{A}^{\rm ss}$ has fibre $(z^2+w^2+g(x,y)=0)$ over each $g\in \mathbf{A}^{\rm ss}$.
	By the ``if'' part we know that each fibre of $\mathcal{X}\to \mathbf{A}^{\rm ss}$ is K-semistable. 

	Consider the $(\bG_m\times \mathrm{SL}_2)$-action $\lambda$ on $\mathbf{A}$ given by 	$\lambda(t,A)\cdot g(x,y)=t^2 g(A^{-1}(x,y))$. It is clear that $\mathbf{A}^{\rm ss}$ is a $(\bG_m\times\mathrm{SL}_2)$-invariant open subset. Then there is a $(\bG_m\times\mathrm{SL}_2)$-action $\tilde{\lambda}$ on $\mathcal{X}$ as a lifting of $\lambda$ given by 
	\[
	 \tilde{\lambda}(t,A)\cdot ([x,y,z,w],g):=([A(x,y),tz,tw], \lambda(t,A)\cdot g).
	\]
    Denote by $\mathcal{M}^{\mathrm{GIT}}:=[\mathbf{A}^{\rm ss}/(\bG_m\times\mathrm{SL}_2)]$ and $M^{\mathrm{GIT}}:=\mathbf{P}\mathbin{/\mkern-6mu/} \mathrm{SL}_2$ where $\mathbf{P}:=\bP(\mathbf{A})$. It is clear that $M^{\mathrm{GIT}}$ is the good moduli space of $\mathcal{M}^{\mathrm{GIT}}$. Taking quotient of the family $\mathcal{X}\to\mathbf{A}^{\rm ss}$ by $
    \tilde{\lambda}$, we obtain a $\mathbb{Q}$-Gorenstein flat family of K-semistable $\mathbb{Q}$-Fano varieties over $\mathcal{M}^{\mathrm{GIT}}$, where fibres over closed points are precisely K-polystable fibres. 
    
    From a series of important recent works \cite{Jia17, LWX18, CP18, blum_xu_uniqueness, ABHLX19, Xu19, BLX19, XZ19, XZ20, BHLLX20, LXZ21}, we know that there exists an Artin stack of finite type $\mathcal{M}_{2,8/a}^{\rm Kss}$ parametrizing K-semistable (possibly singular) del Pezzo surfaces of degree $8/a$. Moreover, $\mathcal{M}_{2,8/a}^{\rm Kss}$ admits a projective good moduli space $M_{2,8/a}^{\rm Kps}$ parametrizing K-polystable ones. Let $\mathcal{M}^{\mathrm{K}}$ be the Zariski closure (with reduced structure) of the locally closed substack in $\mathcal{M}_{2,8/a}^{\rm Kss}$ parametrizing K-semistable degree $2a$ weighted hypersurfaces $X\subset \bP(1,1,a,a)$. Let $M^{\mathrm{K}}$ be the good moduli space of $\mathcal{M}^{\mathrm{K}}$ as a closed  algebraic subspace of $M_{2,8/a}^{\rm Kps}$. 
    Then the above construction and the ``if'' part produces a morphism $\Phi:\mathcal{M}^{\mathrm{GIT}}\to \mathcal{M}^{\mathrm{K}}$ which descends to a morphism $\phi:M^{\mathrm{GIT}}\to M^{\mathrm{K}}$. Since a general weighted hypersurface $X$ has the form $z^2+w^2+g(x,y)=0$ in a suitable coordinate where $g\neq 0$ has no multiple linear factors, we know that $\Phi$ is dominant. The ``if'' part shows that $\Phi$ sends closed points to closed points. Since $M^{\mathrm{GIT}}$ is projective, we know that $\phi$ is proper and dominant, which implies that $\phi$ is surjective. Moreover, since $\mathrm{SL}_2$ has no non-trivial characters, we have injections
    \[
     \mathrm{Pic}(M^{\mathrm{GIT}})=\mathrm{Pic}(\mathbf{P}\mathbin{/\mkern-6mu/}\mathrm{SL}_2)\hookrightarrow \mathrm{Pic}_{\mathrm{SL}_2}(\mathbf{P}^{\rm ss})\hookrightarrow \mathrm{Pic}(\mathbf{P}^{\rm ss})
    \]
    by \cite[Proposition 4.2 and Section 2.1]{KKV89}. It is clear that $\mathbf{P}\setminus\mathbf{P}^{\rm ss}$ has codimension at least $2$ in $\mathbf{P}$. Thus we have $\mathrm{Pic}(\mathbf{P}^{\rm ss})\cong \mathrm{Pic}(\mathbf{P})\cong \mathbb{Z}$. In particular, the GIT quotient $M^{\mathrm{GIT}}$ has Picard rank $1$. It is clear that $M^{\mathrm{K}}$ is not a single point. Thus $\phi:M^{\mathrm{GIT}}\to M^{\mathrm{K}}$ is a finite surjective morphism by Zariski's main theorem.
    
    Next, we show that K-poly/semistability implies GIT poly/semistability. Since $\phi$ is surjective,  a K-polystable hypersurface $X\subset\bP(1,1,a,a)$ satisfies that $[X]=\phi([g])\in M^{\mathrm{K}}$ for some GIT polystable binary form $g\in \mathbf{A}\setminus\{0\}$. Thus $X$ has the form $z^2+w^2+g(x,y)=0$ with $g\neq 0$ being GIT polystable. If $X\subset\bP(1,1,a,a)$ is K-semistable, then it specially degenerates to a K-polystable point $[X_0]\in M^{\mathrm{K}}$ by \cite{LWX18}. Clearly $X_0$ has the form $z^2+w^2+g_0(x,y)=0$ with $g_0\neq 0$ being GIT polystable. Since the rank of quadratic forms cannot jump up under degeneration, the quadratic terms in $(z,w)$ of the equation of $X$ has rank $2$, which implies that $X=(z^2+w^2+g(x,y)=0)$ for some $g$. By \cite[Corollary 1.7]{Fuj19}, we know that $(\bP(1,1,a), \frac{1}{2}D)$ is K-semistable where $D=(z^2+g(x,y)=0)$. Since $X$ carries a $\mathbb{Z}/2\mathbb{Z}$-action given by $w\mapsto -w$, we may assume that the special degeneration from $X$ to $X_0$ is $\mathbb{Z}/2\mathbb{Z}$-equivariant by \cite{LZ20, Zhu20}. In particular, this shows that $(\bP(1,1,a), \frac{1}{2}D)$ specially degenerates to $(\bP(1,1,a), \frac{1}{2}D_0)$ where $D_0=(z^2+g_0(x,y)=0)$. By the lower semi-continuity of lct (see e.g.\ \cite{demailly_kollar}), we know that $\lct(\bP(1,1,a); D)\geq \lct(\bP(1,1,a);D_0)\geq \frac{a+2}{2a}$ where the latter inequality was proven in the ``if'' part due to the fact that $g_0$ is GIT polystable. Thus this shows that $g\neq 0$, and each linear factor in $g(x,y)$ has multiplicity at most $a$. Thus we obtain the GIT semistability of $g$.
    The proof of the ``only if'' part is finished.
   
	Finally, we show that any hypersurface $X\subset\bP(1,1,a,a)$ of degree $2a$ is not K-stable. If $X$ were K-stable, then it would have equation $z^2 + w^2 + g(x,y) = 0$, or equivalently the equation $zw + g(x,y) = 0$. It is clear that $t \cdot (z,w) = (tz, t^{-1}w)$ defines an effective action of $\mathbb{G}_\mathrm{m}$ on $X$. Thus $X$ is not K-stable by definition.
\end{proof}



	
	\begin{proof}[Proof of Corollary~\ref{cor:k-poly}]		
		It is clear that $X$ is quasi-smooth if and only if, up to an automorphism of $\PP(1,1,a,a)$, $X$ has the equation $z^2+w^2+g(x,y)=0$ where $g$ has no multiple linear factors. Thus by Theorem~\ref{thm:k=git}  we conclude that $X$ is K-polystable and not K-stable. The existence of KE metrics on $X$ follows from \cite{ltw_1}.
	\end{proof}
	
	\begin{remark}\label{rem:a=2or4}
		For $a=2$, the del Pezzo surface $X$ admits an embedding into $\bP^4$ as a complete intersection of two hyperquadrics. This is induced by the linear system $|-K_X|$ which  is very ample.
		
		For $a=4$, $X$ (as a double cover of $\bP(1,1,4)$) appeared in \cite{odaka_spotti_sun} where it lies in the exceptional divisor of Kirwan blow-up of the GIT moduli space. Hence $X$ admits a $\mathbb{Q}$-Gorenstein smoothing to degree $2$ smooth del Pezzo surfaces.
		
		Therefore, in both cases ($a=2$ or $a=4$) our K-moduli space $M^{\mathrm{K}}$, introduced in the proof of Theorem~\ref{thm:k=git}, form a divisor in the K-moduli spaces of $\mathbb{Q}$-Gorenstein smoothable del Pezzo surfaces of degree $\frac{8}{a}$ studied in \cite{MM93, odaka_spotti_sun}.
		We will see in Proposition~\ref{prop:a=3or>=5} what happens for $a=3$ or $a \geq 5$. 
	\end{remark}



\begin{proposition} \label{prop:a=3or>=5}
	If $a=3$ or $a \geq 5$, then the locus of K-polystable degree $2a$ hypersurfaces in $\PP(1,1,a,a)$ is a connected component of
	$M_{2,8/a}^{\rm Kps}$.
\end{proposition}

\begin{proof}
We denote by $\Gamma$ the connected component of $M_{2,8/a}^{\rm Kps}$ containing K-polystable degree $2a$ hypersurfaces in $\PP(1,1,a,a)$.
In the proof of Theorem~\ref{thm:k=git} we showed that the locus of K-polystable degree $2a$ hypersurfaces in $\PP(1,1,a,a)$ is closed in $\Gamma$; this locus is denoted by $M^{\mathrm{K}}$.
We need to prove that  $M^{\mathrm{K}}$ coincides with $\Gamma$. We will achieve this by a dimension count. Using the notation of the proof of Theorem~\ref{thm:k=git}, there is a finite surjective morphism $\phi: M^{\mathrm{GIT}}\to M^{\mathrm{K}}$. Thus we have
\[
\dim M^{\mathrm{K}} = \dim M^{\mathrm{GIT}} = \dim \mathbf{P} - \dim \mathrm{SL}_2 = 2a-3.
\]

Let us now compute the dimension of $\Gamma$ by analysing the deformation theory of the K-polystable toric del Pezzo surface $Y$ introduced in Proposition~\ref{prop:toric}. Note that a similar study was discussed in \cite{MGS21}.

\medskip

Let $\cT^0_Y$ denote the sheaf of derivations on $Y$, i.e.\ the dual of $\Omega_Y^1$.
Let $\cTqG{1}{Y}$ denote the sheaf of $1$st order $\QQ$-Gorenstein deformations of $Y$.
The singular locus of $Y$, which consists of $4$ points, contains the set-theoretic support of $\cTqG{1}{Y}$.

Since $Y$ is a toric Fano, we have $H^1(\cT^0_Y) = H^2(\cT^0_Y) = 0$ by \cite[\S4.3]{petracci_survey}. Via a standard argument about the local-to-global spectral sequence for Ext, we deduce that the tangent space of the $\QQ$-Gorenstein deformation functor of $Y$ is $H^0(\cTqG{1}{Y})$.
The $\QQ$-Gorenstein deformation functor of $Y$ is unobstructed because $Y$ is a del Pezzo surface with cyclic quotient singularities \cite[Lemma~6]{procams}. Therefore the germ at the origin of the vector space $H^0(\cTqG{1}{Y})$ is the base of the miniversal (Kuranishi) $\QQ$-Gorenstein deformation of $Y$.

Consider the torus $T_N = N \otimes_\ZZ \mathbb{G}_\mathrm{m}$ acting on the toric variety $Y$.
There is an action of $T_N$ on the vector space $H^0(\cTqG{1}{Y})$, hence $H^0(\cTqG{1}{Y})$ splits into the direct sum of irreducible representations (characters) of the torus $T_N$.

We observe that the singularities of $Y$ are:
\begin{itemize}
	\item $2$ points of type $\frac{1}{a}(1,-1) = A_{a-1}$, which correspond to the cones in $\Sigma$ spanned by
	\[
	\pm \begin{pmatrix}
	a \\ 1
	\end{pmatrix},
	\
	\pm \begin{pmatrix}
	0 \\ 1
	\end{pmatrix},
	\]
	\item $2$ points of type $\frac{1}{a}(1,1)$, which correspond to the cones in $\Sigma$ spanned by
	\[
	\pm \begin{pmatrix}
	a \\ 1
	\end{pmatrix},
	\
	\mp \begin{pmatrix}
	0 \\ 1
	\end{pmatrix}.
	\]
\end{itemize}
Since $a=3$ or $a \geq 5$, the surface singularity $\frac{1}{a}(1,1)$ is $\QQ$-Gorenstein rigid, so it does not contribute to $H^0(\cTqG{1}{Y})$. One can see that the $T_N$-representation $H^0(\cTqG{1}{Y})$ is the direct sum of the $1$-dimensional representation of $T_N$ associated to the characters
\begin{equation} \label{eq:weights}
(0, \pm 2), (0, \pm 3), \dots, (0, \pm a) \in M.
\end{equation}
In particular $\dim H^0(\cTqG{1}{Y}) = 2a-2$, so the base of the miniversal $\QQ$-Gorenstein deformation of $Y$ is a smooth germ of dimension $2a-2$.

Since the weights in \eqref{eq:weights} are contained in a rank $1$ sublattice of $M$, there exists a $1$-dimensional subtorus of $T_N$ which acts trivially on $H^0(\cTqG{1}{Y})$. More precisely one can prove that the affine quotient $H^0(\cTqG{1}{Y}) / T_N$ has dimension $2a-3$.

Since every facet of the polytope $P^\circ$ has no interior lattice points, by \cite[Proposition~2.6]{ask_petracci} the automorphism group of $Y$ is $T_N \rtimes \Aut(P)$, where $\Aut(P) \subseteq \mathrm{GL}(N)$ is the finite group consisting of the lattice automorphisms which keep the polytope $P$ invariant. Since the difference between $T_N$ and $\Aut(Y)$ is just a finite group, we deduce that the affine quotient the affine quotient $H^0(\cTqG{1}{Y}) / \Aut(Y)$ has dimension $2a-3$.
By the local structure of the K-moduli space \cite{ABHLX19, luna_etale_slice_stacks} we know that the completion of the local ring of $\Gamma$ at $[Y]$ coincides with the completion at the origin of $H^0(\cTqG{1}{Y}) / \Aut(Y)$. This proves that $\Gamma$ has dimension $2a-3$ at $[Y]$. Since $\dim M^{\mathrm{K}}=2a-3$, we know that $M^{\mathrm{K}}$ is an irreducible component of $\Gamma$. 

Moreover, since all K-polystable del Pezzo surfaces in $M^{\mathrm{K}}$ have cyclic quotient singularities by Theorem \ref{thm:k=git}, they have unobstructed $\mathbb{Q}$-Gorenstein deformations by \cite[Lemma~6]{procams}. Thus the stack $\mathcal{M}_{2,\frac{8}{a}}^{\rm Kss}$ is smooth in an open neighbourhood of $\mathcal{M}^{\mathrm{K}}$. In particular, this implies that $\Gamma$ is normal in an open neighbourhood of $M^{\mathrm{K}}$. Since $M^{\mathrm{K}}$ is an irreducible component of $\Gamma$, we have $M^\mathrm{K} = \Gamma$.
\end{proof}



\begin{comment}
The K-(semi/poly)stability of small $\QQ$-Gorenstein deformations of $Y$ is equivalent to the GIT-(semi/poly)stability of the $\Aut(Y)$-action on $\DefqG{Y}$ \cite[\S2-3]{odaka_spotti_sun}.
In purely algebraic terms this equivalence is the local structure of the K-moduli stack of K-semistable Fano varieties and of the K-moduli space of K-polystable Fano varieties
. From what has been seen above we deduce (ii).

\medskip

It remains to prove (iii). 

This easily implies that the stabiliser of every point in $\DefqG{Y}$ with respect to the action of $\Aut(Y)$ contains $\mathbb{G}_\mathrm{m}$.
By the considerations explained above about the local structure of the K-moduli stack at $Y$, we deduce that the automorphism group of every
small $\QQ$-Gorenstein deformation of $Y$ contains $\mathbb{G}_\mathrm{m}$.




$Y$ is a K-polystable del Pezzo surface with volume $K^2_Y = \frac{8}{a}$.

$Y$ has four singular points: two of type $\frac{1}{a}(1,a-1) = A_{a-1}$ and two of type $\frac{1}{a}(1,1)$.

The base of the miniversal $\QQ$-Gorenstein deformation of $Y$ is smooth of dimension
\[
\begin{cases}
4 &\qquad \text{if } a = 2, \\
8 &\qquad \text{if } a = 4, \\
2a-2 &\qquad \text{if } a=3 \text{ or } a \geq 5. \\	
\end{cases}
\]

The sheaf $\cTqG{1}{Y}$ is supported on the singular locus of $Y$: at the singularity $\frac{1}{a}(1,-1)$ it has length $a-1$ and at the singularity $\frac{1}{a}(1,1)$ it has length
\[
\begin{cases}
1 &\qquad \text{if } a = 2 \text{ or } a=4, \\
0 &\qquad \text{otherwise.}
\end{cases}
\]
This implies
\[
\dim \rH^0(\cTqG{1}{Y}) = \begin{cases}
4 &\qquad \text{if } a = 2, \\
8 &\qquad \text{if } a = 4, \\
2a-2 &\qquad \text{if } a=3 \text{ or } a \geq 5. \\	
\end{cases}
\]



If $a \geq 3$, then the automorphism group of the polytope $P_a$ is the following non-cyclic group of order $4$:
\[
\Aut(P_a) = \left\{ 
\begin{pmatrix}
1 & 0 \\ 0 & 1
\end{pmatrix}
, \
\begin{pmatrix}
-1 & 0 \\ 0 & -1
\end{pmatrix}
, \
\begin{pmatrix}
-1 & a \\ 0 & 1
\end{pmatrix}
, \
\begin{pmatrix}
1 & -a \\ 0 & -1
\end{pmatrix}
\right\} \subset \mathrm{GL}_2(\ZZ)
\]
\end{comment}
		


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