\section*{Introduction}
Throughout the paper, we work with algebraic varieties and
algebraic groups over the field of complex numbers $\mathbb{C}$.

Let $Y$ be a normal irreducible variety whose divisor
class group $\Cl(Y)$ is finitely generated and
$\Gamma(Y, \mathcal{O}_Y^*) = \mathbb{C}^*$. The \em Cox ring \em of $Y$
is the $\Cl(Y)$-graded $\mathbb{C}$-algebra 
\begin{align*}
\mathcal{R}(Y) \coloneqq \bigoplus_{[D] \in \Cl(Y)} \Gamma(Y, \mathcal{O}_Y(D))\text{,}
\end{align*}
with the multiplication defined by the canonical maps
\begin{align*}
\Gamma(Y, \mathcal{O}_Y(D_1)) \otimes \Gamma(Y, \mathcal{O}_Y(D_2)) \to \Gamma(Y, \mathcal{O}_Y(D_1+D_2))\text{.}
\end{align*}
Some accuracy is required in order for this
multiplication to be well-defined
(cf.~\cite{ha} or \cite{coxrings} for details).

It has been shown by Cox that $\mathcal{R}(Y)$ is a polynomial
ring if $Y$ is a toric variety (cf.~\cite{tcox}). The
converse was obtained by Hu and Keel for smooth
projective varieties (cf.~\cite[Corollary~2.10]{huke}).
Toric varieties can be considered
as examples in the more general class of spherical
varieties, which are quasihomogeneous with respect
to the action of a connected reductive group $G$.

The aim of this paper is to investigate the Cox ring of an arbitrary
spherical variety.

We recall some standard notions from the theory of
spherical varieties. 
Let $G$ be a connected reductive group.
A closed subgroup $H \subseteq G$ is called
\em spherical \em if a Borel subgroup $B \subseteq G$
has an open orbit in $G/H$. Then $G/H$
is called a \em spherical homogeneous space\em.
In this case, we may always assume that $BH$ is open in $G$.
A $G$-equivariant open
embedding $G/H \hookrightarrow Y$ into a normal irreducible $G$-variety $Y$
is called a \em spherical embedding\em, and $Y$ is called a \em spherical variety\em.
Let $p: G' \to G$ be a finite covering
such that $G'$ is of simply connected 
type, i.e.~$G' = G^{ss} \times C$ where $G^{ss}$ is semisimple simply
connected and $C$ is a torus. Then $G/H \cong G'/p^{-1}(H)$,
and these two spherical homogeneous spaces have exactly the same embeddings.
Therefore in this paper we will always assume $G = G^{ss} \times C$.
Similarly to the theory of toric varieties, spherical embeddings
$G/H \hookrightarrow Y$ can be described by some combinatorial
data introduced by Luna and Vust (cf.~\cite{lunavust} and \cite{knopsph}).

We denote by $\mathcal{M}$ the weight lattice of $B$-eigenvectors in the function
field $\mathbb{C}(G/H)$ and by $\mathcal{N} \coloneqq \Hom(\mathcal{M}, \mathbb{Z})$ the dual lattice
with the natural pairing $\langle \cdot , \cdot \rangle: \mathcal{N} \times \mathcal{M} \to \mathbb{Z}$.
If $\nu: \mathbb{C}(G/H) \to \mathbb{Q}$ is a discrete valuation, its
restriction to $B$-eigenvectors induces a map $u: \mathcal{M} \to \mathbb{Q}$,
which lies in the vector space $\mathcal{N}_\mathbb{Q} \coloneqq \mathcal{N} \otimes_\mathbb{Z} \mathbb{Q}$.
We denote the set of $G$-invariant discrete valuations on $\mathbb{C}(G/H)$ by $\mathcal{V}$.
The above assignment is injective on $\mathcal{V}$ and therefore defines an inclusion
$\mathcal{V} \subseteq \mathcal{N}_\mathbb{Q}$. The set $\mathcal{V}$ is a polyhedral convex cone called the \em valuation
cone \em of $G/H$. In fact, the cone $\mathcal{V}$ is the intersection of half-spaces
$\{u \in \mathcal{N}_\mathbb{Q} : \langle u, \gamma_i \rangle \le 0\}$ where
$\gamma_1, \ldots, \gamma_s$ are linearly independent primitive elements in the
lattice $\mathcal{M}$ called the \em spherical roots \em
of $G/H$.

A spherical embedding $G/H \hookrightarrow Y$ is called a \em wonderful
completion \em if $Y$ is complete, smooth, and contains exactly one closed $G$-orbit.
Then $Y$ is called a \em wonderful variety\em.
A wonderful completion of $G/H$ exists (and is unique) if and only if the valuation cone $\mathcal{V}$ is spanned by a
basis of $\mathcal{N}$. 
To any spherical homogeneous space $G/H$ one can associate in a natural way
a closed subgroup $H' \times C \subseteq N_G(H)$ of finite index and containing $H$
such that a wonderful completion $G/(H' \times C) \cong G^{ss}/H' \hookrightarrow Y'$ exists (cf.~\cite{Luna:typea}).

Let $G/H \hookrightarrow Y$
be a spherical embedding.
As the Cox ring does not depend on $G$-orbits of codimension
two or greater, we may assume that $Y$ contains only non-open
$G$-orbits of codimension one, which are exactly the
$G$-invariant prime divisors $Y_1, \ldots, Y_n$ in $Y$. To each $Y_l$
we assign the primitive element $u_l \in \mathcal{N}$ corresponding
to the discrete valuation on $\mathbb{C}(G/H)$ induced by $Y_l$.
It follows from the Luna-Vust theory that the embedding $G/H \hookrightarrow Y$
can be described combinatorially by the fan $\Sigma$ in $\mathcal{N}_\mathbb{Q}$ 
consisting exactly of the trivial cone $0$ and the one-dimensional cones $\sigma_l \coloneqq \mathbb{Q}_{\ge 0}u_l$,
which lie in the valuation cone $\mathcal{V}$.

The Cox ring of $Y$
has been described by Brion (cf.~\cite{brcox}).
In fact, he describes the equivariant Cox ring $\mathcal{R}_G(Y) \cong \mathcal{R}(Y) \otimes_\mathbb{C} \mathbb{C}[C]$.
As a first step, he computes the Cox ring $\mathcal{R}(Y')$ of the associated wonderful completion $G^{ss}/H' \hookrightarrow Y'$
as the Rees algebra of a certain filtration of the ring $\glob{G^{ss}/K'}$ where $K'$ is
the intersection of the kernels of all characters of $H'$.
The equivariant Cox ring $\mathcal{R}_G(Y)$ is then obtained from $\mathcal{R}(Y')$ by a base change.
There is a natural bijection between the set $\{\gamma_1, \ldots, \gamma_s\}$
of spherical roots of $G^{ss}/H'$ and the set $\{Y'_1, \ldots, Y'_s\}$ of
$G$-invariant prime divisors in $Y'$.
It immediately follows from the result of Brion that
\begin{align*}
\mathcal{R}(Y) \cong \mathcal{R}(Y') \otimes_{\mathbb{C}[Z_1, \ldots, Z_s]} \mathbb{C}[W_1, \ldots, W_n]
\end{align*}
where $\mathbb{C}[Z_1, \ldots, Z_s]$ and $\mathbb{C}[W_1, \ldots, W_n]$ are polynomial rings.
The homomorphism $\mathbb{C}[Z_1, \ldots, Z_s] \to \mathcal{R}(Y')$ sends
$Z_i$ to the canonical section in $\Gamma(Y', \mathcal{O}_{Y'}(Y'_i))$,
and the homomorphism $\mathbb{C}[Z_1, \ldots, Z_s] \to \mathbb{C}[W_1, \ldots, W_n]$
sends
\begin{align*}
Z_i \mapsto \prod_{l=1}^n W_l^{-\langle u_l, \gamma_i \rangle}\text{.}
\end{align*}

Our main result is another description of the Cox ring $\mathcal{R}(Y)$.
We will reduce the general case
to the case where the spherical homogeneous space $G/H$
has trivial divisor class group and is therefore quasiaffine.
We show that the Cox ring $\mathcal{R}(Y)$ is obtained by homogenizing the
equations of the affine closure of $G/H$ inside an appropriately constructed 
embedding $G/H \hookrightarrow \mathbb{C}^d \times (\mathbb{C}^*)^m$.
As a byproduct,
we obtain a description of the valuation cone $\mathcal{V}$
in terms of tropical algebraic geometry.
We will compare our results with the approach
of Brion in Section~\ref{five}.

We now give a detailed summary of our results. 
We assume that the homogeneous space $G/H$ has
trivial divisor class group.
Let $\mathcal{D} \coloneqq \{D_1, \ldots, D_r\}$ be the set of $B$-invariant
prime divisors in $G/H$, and assume that the spherical embedding $G/H \hookrightarrow Y$
satisfies $\Gamma(Y, \mathcal{O}_Y^*) = \mathbb{C}^*$.
We choose prime elements $f_1, \ldots, f_r \in \Gamma(G/H, \mathcal{O}_{G/H})$
with $\mathbb{V}(f_i) = D_i$ and obtain irreducible $G$-modules $V_i \coloneqq \langle G \cdot f_i
\rangle \subseteq \Gamma(G/H, \mathcal{O}_{G/H})$.
For each $i$ we set $s_i \coloneqq \dim V_i$ and choose a basis $\{f_{ij}\}_{j=1}^{s_i} \subseteq
G \cdot f_i$ of $V_i$ with $f_{i1} = f_i$.
We let $m$ be the rank of the finitely generated free abelian group
$\Gamma(G/H, \mathcal{O}^*_{G/H})/\mathbb{C}^* \cong \mathfrak{X}(G)^H \cong \mathfrak{X}(C)^{p_2(H)}$ where $p_2: G \to C$ denotes the projection
and choose representatives $\{g_k\}_{k=1}^m \subseteq \Gamma(G/H, \mathcal{O}^*_{G/H})$ of a basis.
The $B$-weights of $f_1, \ldots, f_r$ and $g_1, \ldots, g_m$ freely generate lattices $\mathcal{M}_V$ and $\mathcal{M}_T$ respectively, and
we obtain $\mathcal{M} = \mathcal{M}_V \oplus \mathcal{M}_T \cong \mathbb{Z}^{r+m}$.
Finally, we define the torus $T \coloneqq C/p_2(H) \cong \Spec(\mathbb{C}[\mathcal{M}_T])$.

The next step is to define a $G$-equivariant locally closed embedding
\begin{align*}
G/H \hookrightarrow Z \coloneqq V^* \times T \cong \mathbb{C}^{s_1+\ldots+s_r} \times (\mathbb{C}^*)^m
\end{align*}
where $V^*$ is dual to $V \coloneqq V_1 \oplus \ldots \oplus V_r$.
We have $\mathbb{C}[Z] = S(V) \otimes_\mathbb{C} \mathbb{C}[\mathcal{M}_T]$, where
$S(V)$ denotes the symmetric algebra of $V$.
The coordinate ring of $V_i^*$ is the symmetric algebra $S(V_i)$, whose
generators corresponding to the above basis we denote by $S_{ij}$ for $1 \le j \le s_i$,
i.e.~$\mathbb{C}[V_i^*] = S(V_i) = \mathbb{C}[S_{i1}, \ldots, S_{is_i}]$.
We denote the generators of the coordinate ring of $T$ corresponding to the
above basis of $\mathcal{M}_T$ by $T_k$ for $1 \le k \le m$, i.e.~$\mathbb{C}[T] = \mathbb{C}[\mathcal{M}_T] = \mathbb{C}[T_1^{\pm 1}, \ldots,
T_m^{\pm 1}]$.
The locally closed embedding $G/H \hookrightarrow Z$ is then given
by the $G$-equivariant surjective map $\mathbb{C}[Z] \to \Gamma(G/H, \mathcal{O}_{G/H})$
sending
$S_{ij} \mapsto f_{ij}$ and
$T_k \mapsto g_k$.
Its kernel is the prime ideal $\mathbb{I}({G/H})$.
Considering the natural action of the torus $\Spec(\mathbb{C}[\mathcal{M}])$ on $Z$, we obtain
a corresponding $\mathcal{M}$-grading on the coordinate ring $\mathbb{C}[Z]$. For $f \in \mathbb{C}[Z]$ and
$\mu \in \mathcal{M}$ we denote the $\mu$-homogeneous
component of $f$ by $f^{(\mu)}$.

In order to describe the relations of $\mathcal{R}(Y)$, we define a homogenization operation
in two steps. The first step is the map $\alpha: \mathbb{C}[Z] \to \big(\mathbb{C}[Z]\big)[W_1, \ldots, W_n]$
defined as follows.
For each $f \in \mathbb{C}[Z]$ and $u \in \mathcal{N}$ we define
\begin{align*}
{\rm ord}_u(f) \coloneqq \min_{\mu\in\mathcal{M}} \mathopen{}\mathclose\bgroup\left\{\mathopen{}\mathclose\bgroup\left\langle u, \mu \aftergroup\egroup\right\rangle; f^{(\mu)} \ne 0\aftergroup\egroup\right\}\text{,}
\end{align*}
and set
\begin{align*}
f^\alpha \coloneqq \frac{\sum_{\mu \in \mathcal{M}} \mathopen{}\mathclose\bgroup\left(f^{(\mu)} \prod_{l=1}^n W_l^{\mathopen{}\mathclose\bgroup\left\langle u_l, \mu \aftergroup\egroup\right\rangle}\aftergroup\egroup\right)}
{\prod_{l=1}^n W_l^{{\rm ord}_{u_l}(f)}}\text{.}
\end{align*}
The second step is the map $\beta: \big(\mathbb{C}[Z]\big)[W_1, \ldots, W_n]
\to S(V)[W_1, \ldots, W_n]$ sending
$T_k \mapsto 1$ for each $1 \le k \le m$.
Finally, we define the map $h: \mathbb{C}[Z] \to S(V)[W_1, \ldots, W_n]$
by composing the two steps, i.e.~$h \coloneqq \beta \circ \alpha$.
Note that we write the application of the maps $\alpha$, $\beta$, and $h$
as exponents, for example we write $f^h$ instead of $h(f)$.
We can now state our main result.

\begin{mt}
We have
\begin{align*}
\mathcal{R}(Y) \cong S(V)[W_1, \ldots, W_n]\mathopen{}\mathclose\bgroup\left/\middle(f^h; f \in \mathbb{I}(G/H)\aftergroup\egroup\right)\text{,}
\end{align*}
with $\Cl(Y)$-grading given by $\deg(S_{ij}) = [D_i]$ and $\deg(W_l) = [Y_l]$.
If $H$ is connected, $\mathcal{R}(Y)$ is a factorial ring.
\end{mt}

In particular, if $\mathbb{I}(G/H) = (f)$ is a principal ideal generated
by $f$, then the ideal of relations of $\mathcal{R}(Y)$ is generated by $f^h$.

In the special case of a toric variety $Y$, the Main Theorem reduces to the result of
Cox that $\mathcal{R}(Y)$ is a polynomial ring with one variable per
$G$-invariant prime divisor because the homogeneous space
has no $B$-invariant prime divisors,
i.e.~$V = \langle 0\rangle$, and $\mathbb{I}(G/H) = (0)$.
In the special case of a horospherical variety $Y$, we will see that
$\mathcal{R}(Y) \cong \mathcal{R}(G/P)[W_1, \ldots, W_n]$,
where $P \coloneqq N_G(H)$,
i.e.~the Cox ring of $Y$ is a polynomial
ring over the Cox ring of $G/P$. This also follows
directly from the description of the Cox ring by Brion.

The paper is organized in five sections. 
In Section~\ref{one} we present some 
information about the valuation cone $\mathcal{V}$ and
describe it using tropical algebraic geometry.
The proof of the Main Theorem in the crucial case of a
spherical homogeneous space with trivial divisor class group 
is given in Section~\ref{two}.
Then we explain in Section~\ref{three} how the results can be extended
to arbitrary spherical homogeneous spaces.
In Section~\ref{four}, we illustrate our results by some explicit examples.
Finally, we compare our results with
the approach of Brion in Section~\ref{five}.

\section{The valuation cone $\mathcal{V}$}
\label{one}

We continue to use the notation and the assumptions from the introduction in
this section.

It was shown in \cite{brionsym} and \cite{knopsym} that
the valuation cone $\mathcal{V}$ is a fundamental chamber for the
action of a crystallographic reflection group $W_{G/H}$
on $\mathcal{N}_\mathbb{Q}$ called the \em little Weyl group \em (cf.~\cite[Theorem~22.13]{ti}).
In particular, the cone $\mathcal{V}$ is cosimplicial,
and the set of spherical roots is the
set of simple roots of a root system
with Weyl group $W_{G/H}$.
We have the isotypic decomposition into $G$-modules
\begin{align*}
\glob{G/H} = \bigoplus_{\mu} V_{\mu}\text{,}
\end{align*}
where $\mu$ runs over pairwise distinct elements of $\mathcal{M}$
and $V_{\mu}$ is an irreducible $G$-module of highest weight $\mu$.
Let $\mathcal{T}$ be the cone in $\mathcal{M} \otimes_\mathbb{Z} \mathbb{Q}$
generated by the elements of the form $\mu_1 + \mu_2 - \mu_3$ such that
\begin{align*}
V_{\mu_3} \subseteq V_{\mu_1}\cdot V_{\mu_2}\text{.}
\end{align*}
It follows from \cite[Proposition~2.13]{vbmod} that the cone $\mathcal{T}$
is polyhedral. The cone $\mathcal{T}$ is equal to the cone
generated by the spherical roots,
i.e.~$-\mathcal{V}$ is the dual cone of $\mathcal{T}$ (cf.~\cite[Section~3]{losev} and \cite[Section~5]{knopsph}).

In this sense, the valuation cone $\mathcal{V}$ is related to the failure of
the $\mathbb{C}$-algebra $\glob{G/H}$ being $\mathcal{M}$-graded. 
We have the following extreme case: a spherical homogeneous space
$G/H$ is called \em horospherical \em if $\mathcal{V} = \mathcal{N}_\mathbb{Q}$. In our setting,
this is equivalent to the $\mathbb{C}$-algebra $\glob{G/H}$ being $\mathcal{M}$-graded.
Another characterization of horospherical homogeneous spaces is
that $H$ contains a maximal unipotent subgroup of $G$ (cf.~\cite{pauer2} and \cite[Corollary 6.2]{knopsph}).

We will state a result about the valuation cone which
uses tropical algebraic geometry. An introduction to this subject
can be found in \cite{tropintro}. Let $X$ be the toric variety
associated to a fan $\Sigma_X$ in $N_\mathbb{Q}$ with torus $\mathbb{T}$. If $S \subset \mathbb{T}$ is a closed subset,
the tropicalization $\trop(S) \subseteq N_\mathbb{Q}$ of $S$ is the support of
a polyhedral fan in $N_\mathbb{Q}$. It gives an answer to the question
of which $\mathbb{T}$-orbits in $X$ intersect the closure $\overline{S}$
of $S$ inside $X$:
by a result of Tevelev (cf.~\cite{tev}), the $\mathbb{T}$-orbit
corresponding to $\sigma \in \Sigma_X$ intersects $\overline{S}$ if
and only if $\trop(S)$ intersects $\relint(\sigma)$.

Our plan is to use this machinery on the embedding $G/H \hookrightarrow Z$.
First, we show that this is indeed a locally closed embedding.
Along the way, we also prove some other claims which have been stated in the introduction.

\begin{remark}
By \cite[Proposition 1.3]{knoppic}, 
every $f_i$ is a $B$-eigenvector,
all elements of $\Gamma(G/H, \mathcal{O}^*_{G/H})$ are $G$-eigenvectors, and the
quotient $\Gamma(G/H, \mathcal{O}^*_{G/H})/\mathbb{C}^*$ is a finitely generated free abelian group.
Moreover,
for each $f_i$ the $G$-module spanned by $G \cdot f_i$ in $\glob{G/H}$
is irreducible and finite-dimensional (cf.~\cite[III.1.5]{kraftinv}).
\end{remark}

\begin{prop}
\label{prop-b-eigenvectors}
The $B$-eigenvectors in $\glob{G/H}$ are given by
\begin{align*}
\glob{G/H}^{(B)} = \mathopen{}\mathclose\bgroup\left\{cf_1^{d_1}\cdots f_r^{d_r}; c\in \Gamma(G/H, \mathcal{O}^*_{G/H}), d_i \in \mathbb{N}_0 \aftergroup\egroup\right\}\text{.}
\end{align*}
\end{prop}
\begin{proof}
For each $f \in \glob{G/H}^{(B)}$ all irreducible components of $\mathbb{V}(f)$ are $B$-invariant since
$B$ is irreducible. 
\end{proof}
\begin{prop}
\label{generators}
The $\mathbb{C}$-algebra $\glob{G/H}$ is generated by $\{f_{ij}, g_k^{\pm 1}\}$.
\end{prop}

\begin{proof}
Since every element of $\glob{G/H}^U$ can be written as the sum of $B$-eigenvectors,
the $\mathbb{C}$-algebra $\glob{G/H}^U$ is generated by $\{f_i,g_k^{\pm 1}\}$. By
\cite[III.3.1]{kraftinv}, it follows that $\glob{G/H}$ is generated as claimed.
\end{proof}

For each $f \in \mathbb{C}(G/H)^{(B)}$ we denote by $\chi_f$ its $B$-weight.
For the weight lattice $\mathcal{M}$ of $B$-eigenvectors in $\mathbb{C}(G/H)$ we then have
\begin{align*}
\mathcal{M} &= \{\chi \in \mathfrak{X}(B); \text{there exists } f\in \mathbb{C}(G/H)^{(B)} \text{ with } \chi=\chi_f\}\text{.}
\end{align*}
Recall that there is an exact sequence (cf.~\cite[after 1.7]{knopsph})
\begin{align*}
1 \to \mathbb{C}^* \to \mathbb{C}(G/H)^{(B)} \to \mathcal{M} \to 0\text{.}
\end{align*}
We define $v_i^* \coloneqq \chi_{f_i}$ and $w_k^* \coloneqq \chi_{g_k}$.

\begin{prop}
\label{prop:m-basis}
The lattice $\mathcal{M} \subseteq \mathfrak{X}(B)$ is freely generated by
\begin{align*}
\{v_1^*, \ldots, v_r^*, w_1^*, \ldots, w_m^*\}\text{.}
\end{align*}
\end{prop}
\begin{proof}
Every $f \in \mathbb{C}(G/H)^{(B)}$ can be written as $\frac{g}{h}$ with
$g, h \in \glob{G/H}^{(B)}$ since $\Supp\mathopen{}\mathclose\bgroup\left(\operatorname{div}(f)\aftergroup\egroup\right)$ is
the union of $B$-invariant prime divisors. The claim then follows from
Proposition~\ref{prop-b-eigenvectors}, the exact sequence above,
and the fact that $\{g_k\}_{k=1}^m$ is a basis of $\Gamma(G/H, \mathcal{O}^*_{G/H})/\mathbb{C}^*$.
\end{proof}

We denote the corresponding dual basis of $\mathcal{N}$ by
$\{v_1, \ldots, v_r, w_1, \ldots, w_m\}$. We also have
$\mathcal{M}_V = \langle v_1^*, \ldots, v_r^* \rangle$ and
$\mathcal{M}_T = \langle w_1^*, \ldots, w_m^* \rangle$.

\begin{lemma}
\label{le:alg}
Let $0 \ne h \in \Gamma(G/H, \mathcal{O}_{G/H})$ and $g \in G$
such that $g\cdot h = ch$ where $c$ is a unit. Then we have $c \in \mathbb{C}$.
\end{lemma}
\begin{proof}
We recursively define $h_0 \coloneqq h$ and $h_{i+1} \coloneqq g \cdot h_i$
for $i \in \mathbb{N}_0$. As $h$ is contained in a finite-dimensional $G$-module,
there exists $d \in \mathbb{N}$ such that $\lambda_0h_0 + \ldots +\lambda_d h_d=0$
for some coefficients $\lambda_i \in \mathbb{C}$. As $c$ is a $G$-eigenvector
(cf.~\cite[Proposition 1.3]{knoppic}), dividing by $h$ yields
a polynomial relation for $c$. Hence $c$ is algebraic over $\mathbb{C}$.
\end{proof}

\begin{prop}
The $f_{ij}$ are pairwise nonassociated prime elements,
\end{prop}
\begin{proof}
For every $f_{ij}$ we have $f_{ij} \in G \cdot f_i$ and $f_i$ is prime, hence $f_{ij}$ is also prime.

Now let $f_{i_1j_1} = cf_{i_2j_2}$ for some unit $c$. Then $g\cdot f_{i_1} = c'f_{i_2}$ for some unit $c'$ and $g \in G$.
By Proposition~\ref{prop:m-basis}, we have $i_1 = i_2$, so there exists $g' \in G$
with $g'\cdot f_{i_2j_2} = f_{i_1j_1} = cf_{i_2j_2}$. By Lemma~\ref{le:alg}, it follows that $c \in \mathbb{C}$, therefore $j_1 = j_2$.
\end{proof}

The map defined in the introduction
\begin{align*}
\Phi: S(V) \otimes_\mathbb{C} \mathbb{C}[T] &\to \glob{G/H} \\
S_{ij} &\mapsto f_{ij}\\
T_k &\mapsto g_k\text{,}
\end{align*}
is surjective by Proposition~\ref{generators} and $G$-equivariant, 
hence indeed induces a locally closed $G$-equivariant
embedding $G/H \hookrightarrow Z$
with respect to the corresponding $G$-action on $Z$.

Finally, we have to explain how $Z$ can be naturally regarded as a toric variety.
We denote by $M$ the finitely generated free abelian group with
basis 
\begin{align*}
\{S_{ij}, T_k; 1 \le i \le r, 1 \le j \le s_i, 1 \le k \le m\}\text{,}
\end{align*}
which is isomorphic to $\mathbb{Z}^{s_1+\ldots+s_r+m}$. We define the torus $\mathbb{T} \coloneqq \Spec(\mathbb{C}[M])$ with character lattice $M$, denote the dual lattice by $N \coloneqq \Hom(M, \mathbb{Z})$,
denote the corresponding dual basis by
\begin{align*}
\{v_{ij}, w_k; 1 \le i \le r, 1 \le j \le s_i, 1 \le k \le m\}\text{,}
\end{align*}
 and set $N_\mathbb{Q} \coloneqq N \otimes_\mathbb{Z} \mathbb{Q}$.
The surjective map $M \to \mathcal{M}$ sending $S_{ij} \mapsto v_i^*$ and 
$T_k \mapsto w_k^*$ induces an inclusion $\mathcal{N}_\mathbb{Q} \hookrightarrow N_\mathbb{Q}$
sending $v_i \mapsto v_{i1} + \ldots + v_{is_i}$ and $w_k \mapsto w_k$.
The action of $\mathbb{T}$ on $Z$ makes $Z$ a toric variety.

\begin{theorem}
\label{prop-trop}
We have
\begin{align*}
\mathcal{V} = \trop(G/H \cap \mathbb{T}) \cap \mathcal{N}_\mathbb{Q} \text{.}
\end{align*}
\end{theorem}

The proof will be given at the end of Section~\ref{two}.
We can compare this result to the approach Luna and Vust introduced in
\cite{lunavust} using formal curves (cf.~\cite[Chapter~24]{ti}),
which shows that every $G$-invariant discrete valuation $\nu$
can be obtained up to proportionality by choosing a $\mathbb{C}(\!(t)\!)$-valued point $x(t)$ of $G/H$
and defining
\begin{align*}
\nu(f) \coloneqq {\rm ord}(f(g\cdot x(t)))
\end{align*}
where $g \in G$ is a general point depending on $f \in \mathbb{C}(G/H)$.

On the other hand, taking into account the fundamental theorem of tropical algebraic
geometry (cf.~\cite[Theorem~2.1]{maintrop}), Theorem~\ref{prop-trop}
implies that any $G$-invariant discrete valuation $\nu$ comes from
a $\mathbb{C}\{\!\{t\}\!\}$-valued point $x(t)$ of $G/H$ satisfying
\begin{align*}
{\rm ord}(f_{ij}(x(t))) = {\rm ord}(f_i(x(t)))
\end{align*}
for every $i$ and $j$ and is uniquely determined
by
\begin{align*}
\nu(f_i) = {\rm ord}(f_i(x(t))) \text{ and } \nu(g_k) = {\rm ord}(g_k(x(t)))
\end{align*}
for every $i$ and $k$.

\section{Proof of the Main Theorem}
\label{two}

We continue to use the notation and the assumptions from the previous section in
this section.

\begin{remark}
\label{rem-linear-action}
The action of $G$ on
\begin{align*}
Z \cong \mathbb{C}^{s_1} \times \ldots \times \mathbb{C}^{s_r} \times \mathbb{C}^* \times \ldots \times \mathbb{C}^*
\end{align*}
is linear on each factor.
\end{remark}

The natural action of the torus $\Spec(\mathbb{C}[\mathcal{M}])$ on $Z$ defines a
corresponding $\mathcal{M}$-grading on $\mathbb{C}[Z]$. We set $\mathfrak{p} \coloneqq \mathbb{I}(G/H) = \ker \Phi$.

\begin{prop}
\label{prop-grading}
The prime ideal $\mathfrak{p}$ is $\mathcal{M}$-graded, i.e.\ $\glob{G/H}$ is a $\mathcal{M}$-graded $\mathbb{C}$-algebra,
if and only if $G/H$ is horospherical.
\end{prop}
\begin{proof}
This follows from \cite[Proposition 7.6]{ti}.
\end{proof}

\begin{lemma}
\label{le-stdemb}
Every non-open $G$-orbit of the spherical embedding
\begin{align*}
G/H \hookrightarrow \overline{G/H} \subseteq Z
\end{align*}
is contained in the closure of a $B$-invariant prime divisor in $G/H$.
\end{lemma}
\begin{proof}
It follows from Proposition~\ref{generators} and the definition
of the map $\Phi$ that $G/H$ and $\overline{G/H}$ have the same ring of global sections.
Therefore the non-open $G$-orbits are at least
of codimension two. It follows from the general theory
of spherical embeddings (cf.~\cite{knopsph}) that
any non-open $G$-orbit
lies in the closure of some $B$-invariant
prime divisor in $G/H$ if there is no $G$-invariant
prime divisor.
\end{proof}

\begin{prop}
Let $Z_i \coloneqq \mathbb{V}(S_{ij}; 1 \le j \le s_i) \subseteq Z$.
Then we have
\begin{align*}
G/H = \overline{G/H} \setminus (Z_1 \cup \ldots \cup Z_r)\text{.}
\end{align*}
\end{prop}
\begin{proof}
Let $O$ be a non-open $G$-orbit in $\overline{G/H}$. By Lemma~\ref{le-stdemb},
there exists a $D_i$ with $O \subseteq \overline{D_i}$. It follows that
$S_{i1}$ vanishes on $O$. Since $V_i$
is an irreducible $G$-module and $O$ is $G$-invariant, the whole module
vanishes on $O$, so $O \subseteq Z_i$ follows.
\end{proof}

We set
\begin{align*}
X_0 \coloneqq Z \setminus (Z_1 \cup \ldots \cup Z_r)\text{.}
\end{align*}
Then $G/H \hookrightarrow X_0$ is a closed embedding, and $G$ and $\mathbb{T}$ act on $X_0$.
Consider the fan $\Sigma_{X_0}$ in $N_\mathbb{Q}$ corresponding to the toric variety
$X_0$. Our plan is to define a fan $\Sigma_X$ extending
the fan $\Sigma_{X_0}$ such that the closure of $G/H$
inside the toric variety $X$ associated
to the fan $\Sigma_X$ is the spherical variety $Y$ corresponding to the
fan $\Sigma$.
We have already established the natural inclusion $\mathcal{N}_\mathbb{Q} \hookrightarrow N_\mathbb{Q}$,
so the cones of the fan $\Sigma$ in $\mathcal{N}_\mathbb{Q}$ can as well be considered as cones
in $N_\mathbb{Q}$. But extending $\Sigma_{X_0}$ by these one-dimensional cones
is not enough, since it might be impossible to extend the action of $G$ on $X_0$
to the resulting toric variety $X$. The plan can be carried out, however,
if we also add some higher-dimensional cones.

We construct the fan $\Sigma_X$ in $N_\mathbb{Q}$ as follows.
We first define the set
\begin{align*}
\mathfrak{A} \coloneqq \mathopen{}\mathclose\bgroup\left\{\mathfrak{a} \subseteq \mathopen{}\mathclose\bgroup\left\{v_{ij}\aftergroup\egroup\right\}; \text{for each $i$ there is exactly one $j$ with $v_{ij} \notin \mathfrak{a}$}  \aftergroup\egroup\right\}\text{.}
\end{align*}
For each $1 \le l \le n$ and $\mathfrak{a} \in \mathfrak{A}$ we define the cone
\begin{align*}
\sigma_{l, \mathfrak{a}} \coloneqq \cone\mathopen{}\mathclose\bgroup\left(\mathopen{}\mathclose\bgroup\left\{u_l\aftergroup\egroup\right\} \cup \mathfrak{a}\aftergroup\egroup\right)\subseteq N_\mathbb{Q} \text{} 
\end{align*}
and set
\begin{align*}
\Sigma_X \coloneqq \fan\mathopen{}\mathclose\bgroup\left(\{\sigma_{l,\mathfrak{a}}; 1 \le l \le n, \mathfrak{a} \in \mathfrak{A}\}\aftergroup\egroup\right)\text{,}
\end{align*}
the fan generated by the $\sigma_{l,\mathfrak{a}}$ in $N_\mathbb{Q}$.
\begin{prop}
The cones $\sigma_{l,\mathfrak{a}}$ are smooth and compatible.
In particular, the fan $\Sigma_X$ is well-defined.
\end{prop}
\begin{proof}
Let $\sigma_{l,\mathfrak{a}}$ be a cone as defined above. We define the complement
\begin{align*}
\mathfrak{a}^\mathsf{c} \coloneqq \{v_{ij}, w_k : 1\le i \le r, 1\le j \le s_i, 1\le k \le m\} \setminus \mathfrak{a}\text{.}
\end{align*}
We have $N = \langle \mathfrak{a} \rangle \oplus \langle \mathfrak{a}^\mathsf{c} \rangle$.
We write $u_l = u_l' + u_l''$
with $u_l' \in \langle \mathfrak{a} \rangle$ and $u_l'' \in \langle \mathfrak{a}^\mathsf{c} \rangle$.
Then $u_l''$ is primitive, hence $\sigma_{l,\mathfrak{a}}$ is smooth.

Now let $\sigma_{l',\mathfrak{a}'}$ be another cone as defined above, and
let $v \in \sigma_{l,\mathfrak{a}} \cap \sigma_{l',\mathfrak{a}'}$.
We have
\begin{align*}
v &= \sum_{v_{ij} \in \mathfrak{a} \setminus \mathfrak{a}'} d_{ij}v_{ij}
+ \sum_{v_{ij} \in \mathfrak{a} \cap \mathfrak{a}'} d_{ij}v_{ij} + d_0u_l\\
v &= \sum_{v_{ij} \in \mathfrak{a}' \setminus \mathfrak{a}} d'_{ij}v_{ij}
+ \sum_{v_{ij} \in \mathfrak{a} \cap \mathfrak{a}'} d'_{ij}v_{ij} + d'_0u_{l'}
\end{align*}
for some uniquely determined coefficients $d_{ij}, d'_{ij}, d_0, d'_0 \in \mathbb{Q}^+_0$. 
We define
\begin{align*}
m_i = \min \{ \langle v, v_{ij}^* \rangle : 1 \le j \le s_i\}\text{.}
\end{align*}
As $\langle v, v_{ij}^* \rangle = m_i$ for $v_{ij} \notin \mathfrak{a} \cap \mathfrak{a}'$,
the coefficients in the first sum are zero on both lines. It is then clear that
$d_0 = d'_0 = 0$ if $l \ne l'$. It follows that the intersection of the cones
$\sigma_{l,\mathfrak{a}}$ and $\sigma_{l',\mathfrak{a}'}$ is a face of both.
Hence the cones are compatible.
\end{proof}

We denote by $X$ the toric variety associated to the fan $\Sigma_X$.
As the fan $\Sigma_{X_0}$ can also be obtained
by the above construction from the cones $\sigma_{0,\mathfrak{a}}$ where $u_0 \coloneqq 0$, the fan $\Sigma_X$ extends $\Sigma_{X_0}$.
Hence we have an open embedding $X_0 \subseteq X$.
We will now show that the $G$-action on $X_0$ can
be extended to $X$.
We consider $\widehat{N} \coloneqq N \oplus \mathbb{Z}^n$
and regard $N_\mathbb{Q} \subseteq \widehat{N}_\mathbb{Q} \coloneqq \widehat{N} \otimes_\mathbb{Z} \mathbb{Q}$ as naturally included.
We denote the standard basis of $\mathbb{Z}^n$ by
$\{e_l\}_{l=1}^n$. We set
\begin{align*}
\widehat{\sigma}_{l,\mathfrak{a}} &\coloneqq \cone\mathopen{}\mathclose\bgroup\left(\mathopen{}\mathclose\bgroup\left\{e_l\aftergroup\egroup\right\} \cup \mathfrak{a}\aftergroup\egroup\right) \subseteq \widehat{N}_\mathbb{Q}
\end{align*}
and
\begin{align*}
\Sigma_{\widehat{X}} &\coloneqq \fan\mathopen{}\mathclose\bgroup\left(\{\widehat{\sigma}_{l,\mathfrak{a}};  1 \le l \le n, \mathfrak{a} \in
\mathfrak{A}\}\aftergroup\egroup\right)\text{.}
\end{align*}
The associated quasiaffine toric variety $\widehat{X}$ comes with a natural toric
morphism
\begin{align*}
p: \widehat{X} \to X\text{}
\end{align*}
defined by the lattice map $\widehat{N} \to N$ sending $v_{ij} \mapsto v_{ij}$, $w_k \mapsto w_k$, and $e_l \mapsto u_l$.

\begin{remark}
\label{rem-sw}
We have
\begin{align*}
\widehat{X} \cong V^* \times T \times \mathbb{A}^n \setminus \widehat{S}\text{,}
\end{align*}
where $\widehat{S}$ is a closed subset of codimension at least two. 
According to the theory in
\cite{tq}, the toric morphism $p: \widehat{X} \to X$
is a good quotient for the action of a subtorus
$\Gamma \subseteq \mathbb{T} \times (\mathbb{C}^*)^n$.
It is even a geometric quotient. The subtorus $\Gamma$ has
a natural parametrization
$\kappa: (\mathbb{C}^*)^n \xrightarrow{~\cong~} \Gamma$.
Denoting by $W_l$ for $1 \le l \le n$ the coordinates
of $(\mathbb{C}^*)^n$, the action of $\Gamma$ on $\widehat{X}$
is given by
\begin{align*}
\kappa(t) \cdot v = \mathopen{}\mathclose\bgroup\left(\prod_{l=1}^n W_l(t)^{-\mathopen{}\mathclose\bgroup\left\langle u_l, v_{i}^* \aftergroup\egroup\right\rangle}\aftergroup\egroup\right) \cdot v\text{,}
\end{align*}
on $v \in V_i^*$, similarly, by
\begin{align*}
\kappa(t) \cdot v = \mathopen{}\mathclose\bgroup\left(\prod_{l=1}^n W_l(t)^{-\mathopen{}\mathclose\bgroup\left\langle u_l, w_k^* \aftergroup\egroup\right\rangle}\aftergroup\egroup\right) \cdot v\text{,}
\end{align*}
on $v$ in the $k$-th factor $\mathbb{C}^*$ of $T \cong (\mathbb{C}^*)^m$, and finally by the natural action on $\mathbb{A}^n$.

We let $G$ act linearly on the first factors of $\widehat{X}$
with the same action as in
Remark~\ref{rem-linear-action} and trivially on $\mathbb{A}^n$. Then
the action of $G$ on $\widehat{X}$ commutes with the action
of the torus $\Gamma$ from Remark~\ref{rem-sw}. In particular, we obtain an action of $G$ on $X$.
\end{remark}

\begin{remark}
\label{rem-psi}
The natural inclusion $N_\mathbb{Q} \subseteq \widehat{N}_\mathbb{Q}$ defines a $G$-equivariant toric closed embedding
\begin{align*}
\psi: X_0 \to \widehat{X}\text{,}
\end{align*}
and we obtain the following commutative diagram.
\begin{align*}
\xymatrix{
 & \widehat{X} \ar^{p}[rd] \\
X_0 \ar[ru]^\psi \ar@{^{(}->}[rr] & & X
}
\end{align*}
This shows that the action of $G$ on $X$
is an extension of the action of $G$ on $X_0$.
\end{remark}

We denote by $Y'$ the closure of $G/H$ inside $X$. It will
now take some time to prove that $Y'$ is indeed
the spherical embedding of $G/H$ associated to the
fan $\Sigma$, i.e.~that $Y'$ can be identified with $Y$.

\begin{prop}
\label{prop-bigspherical}
The preimage $p^{-1}(G/H)$ with the action of $G \times \Gamma$ is a spherical homogeneous space isomorphic to $G/H \times \Gamma$
with the natural action of $G \times \Gamma$.
The isomorphism
\begin{align*}
\xymatrix{
G/H \times \Gamma \ar[r]^-{\cong} & p^{-1}(G/H)
}
\end{align*}
is given by $(x, t) \mapsto t \cdot \psi(x)$.
\end{prop}

\begin{proof}
As $p$ is a geometric quotient, it follows
that $p^{-1}(G/H) = \Gamma \cdot \psi(G/H)$.
\end{proof}

\begin{remark}
\label{tear-map}
If we denote by $W_l$ for $1 \le l \le n$ the coordinates of $\mathbb{A}^n$
as well as the coordinates of $\Gamma$ under the parametrization $\kappa$ from Remark~\ref{rem-sw}, we
have $\mathbb{C}[\mathbb{A}^n] = \mathbb{C}[W_1, \ldots, W_n]$ and $\mathbb{C}[\Gamma] = \mathbb{C}[W_1^{\pm 1}, \ldots, W_n^{\pm 1}]$.
Considering
the natural inclusion
\begin{align*}
\xymatrix{
p^{-1}(G/H) \ar@{^{(}->}[r] & V^* \times T \times \mathbb{A}^n\text{,}
}
\end{align*}
the isomorphism of Proposition~\ref{prop-bigspherical} is induced by the map
\begin{align*}
\Psi: S(V) \otimes_\mathbb{C} \mathbb{C}[T] \otimes_\mathbb{C} \mathbb{C}[\mathbb{A}^n] &\to \mathopen{}\mathclose\bgroup\left(S(V) \otimes_\mathbb{C} \mathbb{C}[T]\aftergroup\egroup\right)/\mathfrak{p} \otimes_\mathbb{C} \mathbb{C}[\Gamma] \\
S_{ij} &\mapsto S_{ij} \otimes  \mathopen{}\mathclose\bgroup\left(\prod_{l=1}^n W_l^{-\mathopen{}\mathclose\bgroup\left\langle u_l, v_{i}^* \aftergroup\egroup\right\rangle}\aftergroup\egroup\right)\\
T_{k} &\mapsto T_{k} \otimes \mathopen{}\mathclose\bgroup\left(\prod_{l=1}^n W_l^{-\mathopen{}\mathclose\bgroup\left\langle u_l, w_k^* \aftergroup\egroup\right\rangle}\aftergroup\egroup\right) \\
W_l &\mapsto W_l\text{.}
\end{align*}
\end{remark}

We denote the closure of $p^{-1}(G/H)$ inside $\widehat{X}$ by $\widehat{Y}$.

\begin{prop}
\label{prop-r}
$\widehat{Y}$ is the quasiaffine spherical embedding of $G/H \times \Gamma$
corresponding to the fan $\widehat{\Sigma}$ in $\mathopen{}\mathclose\bgroup\left(\mathcal{N} \oplus \mathbb{Z}^n\aftergroup\egroup\right)_\mathbb{Q}$ which consists of the one-dimensional cones
\begin{align*}
\mathbb{Q}_{\ge 0}(u_l + e_l)
\end{align*}
for $1 \le l \le n$. Furthermore, the ring $\glob{\widehat{Y}}$ is factorial.
\end{prop}
\begin{proof}
Consider the spherical embedding associated to the fan $\widehat{\Sigma}$, which is quasiaffine
with factorial ring of global sections (cf.~\cite[Proposition~4.1.1]{brcox}).
By possibly adding some
orbits
of codimension at least two,
we can make it affine (cf.~\cite[Proof
of Theorem 6.7]{knopsph})
and denote its coordinate ring by $R$. We will show that it is
the closure of $p^{-1}(G/H)$ inside the affine toric variety 
$V^* \times T \times \mathbb{A}^n$.
Consider the following diagram.
\begin{align*}
\text{{\small $
\xymatrix{
S(V) \otimes_\mathbb{C} \mathbb{C}[T] \otimes_\mathbb{C} \mathbb{C}[\mathbb{A}^n] \ar[r] & R \ar@{^{(}->}[d]\\
& \mathopen{}\mathclose\bgroup\left(S(V) \otimes_\mathbb{C} \mathbb{C}[T]\aftergroup\egroup\right)/\mathfrak{p} \otimes_\mathbb{C} \mathbb{C}[\Gamma]
\ar@{=}[r] & \glob{G/H \times \Gamma}
}
$}}
\end{align*}
As the ring $R$ is normal, it consists of all the elements in the
ring of global sections of
$G/H \times \Gamma$ which have
a nonnegative value under the valuations induced by
the $G$-invariant
prime divisors in $\Spec(R)$.
It follows that the image of $\Psi$ (cf.~Remark~\ref{tear-map})
is contained in $R$.

Additionally, all the $B\times \Gamma$-eigenvectors in $R$,
and even the $G \times \Gamma$-modules generated by them, belong
to the image of the horizontal map, so we obtain
surjectivity of the horizontal map with the same argument as
in Proposition~\ref{generators}.
The composition of the horizontal map and the vertical localization is the same
map as in Proposition~\ref{prop-bigspherical}. It follows that we have
$\overline{p^{-1}(G/H)} = \Spec(R)$ inside 
$V^* \times T \times \mathbb{A}^n$. 

We use the same symbols for elements
of $S(V) \otimes_\mathbb{C} \mathbb{C}[T] \otimes_\mathbb{C} \mathbb{C}[\mathbb{A}^n]$
and their images in $R$.
As the valuations induced
by the $G$-invariant prime divisors in $\Spec(R)$
have values of $0$ or $1$ respectively on the 
$W_l$ and of $0$
on the $S_{ij}$, it follows that
the $W_l$ and the $S_{ij}$ 
are pairwise nonassociated prime elements of $R$.
The $G$-orbits which lie in the closure of $D_i \times \Gamma$
are contained in $\mathbb{V}(S_{ij}; 1 \le j \le s_i)$, while
the other $G$-orbits of codimension at least two are contained in $\mathbb{V}(W_{l_1}) \cap \mathbb{V}(W_{l_2})$
for some $l_1 \ne l_2$. 
Therefore intersecting with
$\widehat{X}$, i.e.~removing the set $\widehat{S}$, removes exactly the $G$-orbits
of codimension at least two, and the result follows.
\end{proof}

We will reuse the following fact from the proof
of Proposition~\ref{prop-r}.

\begin{remark}
\label{re-primes}
The $W_l$ and the $S_{ij}$ are pairwise
nonassociated prime elements of $\glob{\widehat{Y}}$.
\end{remark}

\begin{remark}
\label{rem:norm}
The restriction
\begin{align*}
p|_{\widehat{Y}}: \widehat{Y} \to Y'
\end{align*}
is a good geometric quotient. In particular, $Y'$ is normal.
\end{remark}

In the following Lemma~\ref{le-val}, we will assume that $\Sigma$ contains
only one one-dimensional cone $\sigma$. We also allow
the cone $\sigma$ to lie outside the valuation cone $\mathcal{V}$. In that
case, the spherical variety $Y$ is not defined, but the closure
$Y'$ of $G/H$ inside $X$ can still be constructed. 
Remark~\ref{rem:norm} not being available, the
normality of $Y'$ is not certain.
This more general setting will only be required for one direction of the proof
of Theorem~\ref{prop-trop} at the end of this section.

\begin{lemma}
\label{le-val}
Any irreducible component of $Y' \setminus (G/H)$
which intersects the $\mathbb{T}$-orbit in $X$ corresponding to $\sigma$ induces a
$G$-invariant valuation which induces $\sigma$ (after possibly normalizing $Y'$).
\end{lemma}
\begin{proof}
We consider the affine open toric subvariety $U_\sigma$ of $X$ with two
orbits, the torus and the orbit corresponding to the cone $\sigma$.
Let $\pi \in \mathbb{C}[U_\sigma]$ be the prime element such that
$\mathbb{V}(\pi)$ is the orbit corresponding to the cone $\sigma$.
We
obtain the following commutative diagram.
\begin{align*}
\xymatrix{
\mathbb{C}[U_\sigma] \ar@{^{(}->}[rrr]\ar[d]  &&& \mathopen{}\mathclose\bgroup\left(S(V) \otimes_\mathbb{C} \mathbb{C}[T]\aftergroup\egroup\right)_{\mathfrak{p}} \ar[d]\\
\mathbb{C}[U_\sigma] \big/ \mathopen{}\mathclose\bgroup\left(\mathfrak{p}\cap\mathbb{C}[U_\sigma]\aftergroup\egroup\right) \ar@{^{(}->}^-{\mathfrak{N}}[r] 
& R_1 \ar@{^{(}->}^-{\mathfrak{L}}[r] & R_2 \ar@{^{(}->}[r] & \mathopen{}\mathclose\bgroup\left(S(V) \otimes_\mathbb{C} \mathbb{C}[T]\aftergroup\egroup\right)_{\mathfrak{p}} \big/ \mathfrak{p}  \ar@{=}[r] & \mathbb{C}\mathopen{}\mathclose\bgroup\left(G/H\aftergroup\egroup\right)
}
\end{align*}
It is sufficient to localize at the prime ideal $\mathfrak{p}$ to get the
inclusion in the first row since the $S_{ij}$ are not in $\mathfrak{p}$.
In the bottom row, $\mathfrak{N}$ denotes normalization, and $\mathfrak{L}$
is localization in such a way that $\mathbb{V}(\pi) = \mathbb{V}(\pi_0)$ in $\Spec(R_2)$ 
where $\pi_0 \in R_2$ is a prime element.
Each $L \in \{S_{ij}, T_k\}$ can be written as
$L = c_1\pi^{d_1}/\pi^{d_2}$ with $c_1 \in \mathbb{C}[U_\sigma]^*$ and $d_1, d_2 \in \mathbb{N}_0$
since there are no $\mathbb{T}$-invariant prime divisors other than $\mathbb{V}(\pi)$ in $U_\sigma$.
Therefore we have $L = c_2\pi_0^{d_3d_1}/\pi_0^{d_3d_2}$ for some
$c_2 \in R_2^*$ and $d_3 \in \mathbb{N}_0$. It follows that $\mathbb{V}(\pi_0)$ induces
the correct cone $\sigma$.
\end{proof}

\begin{prop}
\label{prop-main}
$Y'$ is the spherical
embedding of $G/H$ corresponding to the fan $\Sigma$.
\end{prop}

\begin{proof}
As general embeddings of $G/H$ can be obtained by gluing embeddings
with only one closed orbit, we may assume $n = 1$.
In this case, it is clear that $Y'$ has two $G$-orbits,
and Lemma~\ref{le-val} implies that the closed $G$-orbit induces
the correct $G$-invariant valuation.
\end{proof}

From now on, we will identify $Y'$ and $Y$.
We denote by $X_1, \ldots, X_n$
the $\mathbb{T}$-invariant prime divisors in $X$ corresponding
to $u_1, \ldots, u_n$. Note that $\Gamma(Y, \mathcal{O}^*_Y) = \mathbb{C}^*$
implies $\Gamma(X, \mathcal{O}^*_X) = \mathbb{C}^*$.

We temporarily reuse our notation in a more general setting to
give an overview over the next step.
Let $X$ be a toric $\mathbb{T}$-variety with $\Gamma(X, \mathcal{O}^*_X) = \mathbb{C}^*$ and
$\mathbb{T}$-invariant prime divisors $X_1, \ldots, X_n$.
Then we have the canonical quotient construction
$\pi: \widetilde{X} \to X$, where $\widetilde{X}$ is a quasiaffine
toric variety and $\mathcal{R}(X) \cong \glob{\widetilde{X}}$ (cf.~\cite[Theorem 5.1.11]{cls}).
Let $\iota: Y \hookrightarrow X$ be a closed embedding with
$\Gamma(Y, \mathcal{O}^*_Y) = \mathbb{C}^*$ as well as $X$ and $Y$ smooth. It follows from
the work of Hausen (cf.~\cite{ha}) that $\mathcal{R}(Y) \cong \glob{\pi^{-1}(Y)}$ if
$X_l \cap Y$ is an irreducible hypersurface in $Y$ 
intersecting the $\mathbb{T}$-orbit which is dense in $X_l$ for each $1 \le l \le n$
and the map $\iota^*: \Cl(X) \to \Cl(Y)$ induced by
the pullback of Cartier divisors is an isomorphism.
Such an embedding is called \em neat\em.

We now return to our setting. We identify
the prime divisors $D_i$ in $G/H$ and $\mathbb{V}(S_{ij})$ in $X_0$
with their closures inside $Y$ and $X$ respectively
as long as there is no danger of confusion.

\begin{prop}
\label{prop-neat}
The closed embedding
\begin{align*}
\iota: Y \hookrightarrow X
\end{align*}
is a neat embedding in the sense of \cite[Definition 2.5]{ha}.
\end{prop}

\begin{proof}
Consider the following commutative diagram.
\begin{align*}
\xymatrix{
\widehat{Y} \ar@{^{(}->}^{\widehat{\iota}}[r] \ar[d]_{p} & \widehat{X} \ar[d]^{p} \\
Y \ar@{^{(}->}_{\iota}[r] & X 
}
\end{align*}
Since the diagram commutes, $\iota^{-1}(X_l)$ and $\iota^{-1}(\mathbb{V}(S_{ij}))$ are irreducible for each $i$, $j$, and $l$,
and using Remark~\ref{re-primes} as in the proof
of Theorem~\ref{prop-trop}, we obtain that they intersect the corresponding
$\mathbb{T}$-orbit of codimension one in $X$.

We have a pullback map of Cartier divisors $\iota^*: \Div(X) \to \Div(Y)$, and clearly
$\Supp(\iota^*(X_l)) \subseteq Y_l$ and $\Supp(\iota^*(\mathbb{V}(S_{i1}))) \subseteq D_i$.
Using the diagram again, we see that locally the pullbacks of $X_l$
and $\mathbb{V}(S_{i1})$ are prime divisors. Therefore we have
$\iota^*(X_l) = Y_l$ and $\iota^*(\mathbb{V}(S_{i1})) = D_i$.

Using the explicit
descriptions of
the divisor class groups of toric and spherical varieties
(cf.~\cite[Theorem~4.1.3]{cls} and \cite[Proposition~4.1.1]{brcox}),
we obtain that the induced pullback map
$\iota^*: \Cl(X) \to
\Cl(Y)$
is an isomorphism.
\end{proof}

We are now almost ready to describe the Cox ring $\mathcal{R}(Y)$.
We recall the homogenization operation
from the introduction. The first step is the map $\alpha: \mathbb{C}[Z] \to \big(\mathbb{C}[Z]\big)[W_1, \ldots, W_n]$
defined as follows.
For each $f \in \mathbb{C}[Z]$ and $u \in \mathcal{N}$ we define
\begin{align*}
{\rm ord}_u(f) \coloneqq \min_{\mu\in\mathcal{M}} \mathopen{}\mathclose\bgroup\left\{\mathopen{}\mathclose\bgroup\left\langle u, \mu \aftergroup\egroup\right\rangle; f^{(\mu)} \ne 0\aftergroup\egroup\right\}\text{,}
\end{align*}
and set
\begin{align*}
f^\alpha \coloneqq \frac{\sum_{\mu \in \mathcal{M}} \mathopen{}\mathclose\bgroup\left(f^{(\mu)} \prod_{l=1}^n W_l^{\mathopen{}\mathclose\bgroup\left\langle u_l, \mu \aftergroup\egroup\right\rangle}\aftergroup\egroup\right)}
{\prod_{l=1}^n W_l^{{\rm ord}_{u_l}(f)}}\text{.}
\end{align*}
The second step is the map $\beta: \big(\mathbb{C}[Z]\big)[W_1, \ldots, W_n]
\to S(V)[W_1, \ldots, W_n]$ sending
$T_k \mapsto 1$ for each $1 \le k \le m$.
Finally, we define the map $h: \mathbb{C}[Z] \to S(V)[W_1, \ldots, W_n]$
by composing the two steps, i.e.~$h \coloneqq \beta \circ \alpha$.

\begin{theorem}
\label{th-cox}
We have
\begin{align*}
\mathcal{R}(Y) \cong S(V)[W_1, \ldots, W_n]\mathopen{}\mathclose\bgroup\left/\middle(f^h; f \in \mathfrak{p}\aftergroup\egroup\right)\text{,}
\end{align*}
with $\Cl(Y)$-grading given by $\deg(S_{ij}) = [D_i]$ and $\deg(W_l) = [Y_l]$.
\end{theorem}

\begin{proof}
Let $\pi: \widetilde{X} \to X$ be the canonical quotient construction.
We set $\widetilde{Y} \coloneqq \pi^{-1}(Y)$. By \cite[Construction~5.1.4]{coxrings},
the scheme-theoretic fiber $\widetilde{X} \times_X Y$ is reduced. 
There is a canonical toric closed embedding
$\phi: \widetilde{X} \hookrightarrow \widehat{X}$
such that $\pi = p \circ \phi$.
Note that $\phi^*$
sends $f \mapsto f^\beta$. The assertion now follows from Proposition~\ref{prop-neat} and
\cite[Theorem 2.6]{ha} if we show that the ideal $\widehat{\mathfrak{p}}$
of $\widehat{Y}$ in $\widehat{X}$ is
$\mathfrak{p}^\alpha \coloneqq \mathopen{}\mathclose\bgroup\left(f^\alpha; f \in \mathfrak{p}\aftergroup\egroup\right)$.

We use the map $\psi$ from Remark~\ref{rem-psi}. We have $\widehat{\mathfrak{p}} = \mathbb{I}(\Gamma \cdot \psi(\mathbb{V}(\mathfrak{p})))$. Every $f^\alpha \in \mathfrak{p}^\alpha$ vanishes
on $\psi(\mathbb{V}(\mathfrak{p}))$ since $\psi^*(f^\alpha) = f \in \mathfrak{p}$, and $f^\alpha$ is a $\Gamma$-eigenvector,
so $f^\alpha \in \widehat{\mathfrak{p}}$, and $\mathfrak{p}^\alpha \subseteq \widehat{\mathfrak{p}}$ follows. Now,
let $g \in \widehat{\mathfrak{p}}$. As $\widehat{\mathfrak{p}}$ is a homogeneous ideal with
respect to the $\mathfrak{X}(\Gamma)$-grading, all
homogeneous components $g^{(\xi)}$ are in $\widehat{\mathfrak{p}}$.
It is not difficult to see that
$g^{(\xi)} = \mathopen{}\mathclose\bgroup\left(\psi^*\mathopen{}\mathclose\bgroup\left(g^{(\xi)}\aftergroup\egroup\right)\aftergroup\egroup\right)^\alpha \prod_{l=1}^n W_l^{d_l}$
for some exponents $d_l \in \mathbb{N}_0$. Since $\psi^*\mathopen{}\mathclose\bgroup\left(g^{(\xi)}\aftergroup\egroup\right) \in \mathfrak{p}$, the inclusion
$\widehat{\mathfrak{p}} \subseteq \mathfrak{p}^\alpha$ follows.
\end{proof}

\begin{remark}
If $\Gamma(G/H, \mathcal{O}^*_{G/H}) = \mathbb{C}^*$, we have $\widetilde{X} = \widehat{X}$, and $\phi$ is the identity.
\end{remark}

Cox rings are always factorially graded (cf.~\cite[Theorem~7.3]{bhhom}),
but not factorial in general (cf.~\cite[Example~4.2]{cfact}).
A sufficient condition for the Cox ring to be factorial
is the divisor class group being free (cf.~\cite[Proposition~8.4]{bhhom}, also \cite[Corollary~1.2]{ekw}).
The last part of the Main Theorem also provides a sufficient condition for the Cox ring
of a spherical variety to be factorial.

\begin{theorem}
\label{thm-fact-fact}
If $H$ is connected, $\mathcal{R}(Y)$ is a factorial ring.
\end{theorem}
\begin{proof}
The finitely generated free abelian group $\Gamma(G/H, \mathcal{O}^*_{G/H})/\mathbb{C}^*$
is naturally isomorphic to the subgroup $\mathfrak{X}(C)^H \subseteq \mathfrak{X}(C)$
consisting of $H$-invariant characters. The quotient group
$\mathfrak{X}(C) / \mathfrak{X}(C)^H$ is free 
as $H$ is connected.
Therefore there exists a decomposition $\mathfrak{X}(C) = \mathfrak{X}(C)^H \oplus \mathfrak{X}(C)^\dagger$, and
we can choose the $f_{ij}$ in such a way that $C$ acts on the $f_{ij}$ with characters belonging to $\mathfrak{X}(C)^\dagger$.
It follows that for each $1 \le k \le m$ we have a one-parameter subgroup of $C$ acting nontrivially
only on the variable $T_k$. As $\widehat{Y}$ is $C$-invariant, we obtain $\widehat{Y} \cong \widetilde{Y} \times T$.
Therefore the factoriality of $\mathcal{R}(Y) \cong \glob{\widetilde{Y}}$ follows
from the factoriality of $\glob{\widehat{Y}}$ (cf.~Proposition~\ref{prop-r}).
\end{proof}

Finally, we provide the proof of Theorem~\ref{prop-trop}.
 
\begin{proof}[Proof of Theorem~\ref{prop-trop}]
We have to show $\mathcal{V} = \trop(G/H \cap \mathbb{T}) \cap \mathcal{N}_\mathbb{Q}$.
Using \cite[Lemma 2.2]{tev},
we get the inclusion from the left to the right
using Proposition~\ref{prop-main} if
$Y_l$ intersects
the $\mathbb{T}$-orbit which is dense in $X_l$.
This is the case, since otherwise it would
follow from Remark~\ref{re-primes} that
the codimension of $Y_l$ in $Y$ is at least two.
If the inclusion from the right to the left did not hold,
Lemma~\ref{le-val} would yield a
non-existing $G$-invariant valuation.
\end{proof}

\section{Generalization to arbitrary spherical homogeneous spaces}
\label{three}
We now consider the case where the spherical homogeneous space is allowed
to have nontrivial divisor class group.
Let $\arb{G}$ be a connected reductive group and $\arb{H}
\subseteq \arb{G}$ a spherical subgroup. We may again assume
that $\arb{G}$ is of simply connected type,
i.e.~$\arb{G} = \arb{G}^{ss} \times \arb{C}$
where $\arb{G}^{ss}$ is semisimple simply connected and $\arb{C}$ is a torus.

We fix a Borel subgroup $\arb{B} \subseteq \arb{G}$ such that
the base point $1 \in \arb{G}/\arb{H}$ lies in the open $\arb{B}$-orbit and denote
by $\arb{\mathcal{D}} \coloneqq \mathopen{}\mathclose\bgroup\left\{\arb{D}_1, \ldots, \arb{D}_r\aftergroup\egroup\right\}$ the set of $\arb{B}$-invariant
prime divisors in $\arb{G}/\arb{H}$.
For each $\arb{D}_i$ the pullback under the quotient map
$\arb{G} \to \arb{G}/\arb{H}$ is a divisor
with equation $\arb{f}_i \in \mathbb{C}\mathopen{}\mathclose\bgroup\left[\arb{G}\aftergroup\egroup\right]$ where $\arb{f}_i$ is uniquely determined
by being $\arb{C}$-invariant with respect to the action from the left
and $\arb{f}_i(1) = 1$ (cf.~\cite[4.1]{brcox}).
The group $\arb{H}$ acts from the right on each $\arb{f}_i$ with a character
$\arb{\chi}_i \in \mathfrak{X}\mathopen{}\mathclose\bgroup\left(\arb{H}\aftergroup\egroup\right)$. 

We define
\begin{align*}
G &\coloneqq \arb{G} \times (\mathbb{C}^*)^{\arb{\mathcal{D}}} \\
H &\coloneqq \mathopen{}\mathclose\bgroup\left\{\mathopen{}\mathclose\bgroup\left(h, \arb{\chi}_1(h), \ldots, \arb{\chi}_r(h)\aftergroup\egroup\right); h \in \arb{H}\aftergroup\egroup\right\} \subseteq G\text{,}
\end{align*}
and set $B \coloneqq \arb{B} \times (\mathbb{C}^*)^{\arb{\mathcal{D}}}$.
We have a quotient map $\arb{\pi}: G/H \to \arb{G}/\arb{H}$, which is
a good geometric quotient by the torus $(\mathbb{C}^*)^{\arb{\mathcal{D}}}$.
There is a natural isomorphism $H \cong \arb{H}$, and $G/H$ is a spherical homogeneous space.
The pullbacks of the $\arb{B}$-invariant prime divisors in
$\arb{G}/\arb{H}$ under the quotient map $\arb{\pi}$
are exactly the $B$-invariant prime divisors
$D_1, \ldots, D_r$ in $G/H$.

We denote by $\Pic_{\arb{G}}\mathopen{}\mathclose\bgroup\left(\arb{G}/\arb{H}\aftergroup\egroup\right)$ the group
of isomorphism classes of $\arb{G}$-linearized invertible sheaves
on $\arb{G}/\arb{H}$. The character lattice of $(\mathbb{C}^*)^{\arb{\mathcal{D}}}$
is $\mathbb{Z}^{\arb{\mathcal{D}}}$ with standard basis $\{\eta_1, \ldots, \eta_r\}$.

\begin{prop}
\label{prop-ex-res}
The following diagram of natural maps is commutative, and the top row is an exact sequence.
\begin{align*}
\xymatrix{
0 \ar[r] & \arb{\mathcal{M}} \ar[r] & \mathfrak{X}\mathopen{}\mathclose\bgroup\left(\arb{C}\aftergroup\egroup\right) \oplus \mathbb{Z}^{\arb{\mathcal{D}}} \ar[r] \ar[d]_{\cong} &
\Pic_{\arb{G}}\mathopen{}\mathclose\bgroup\left(\arb{G}/\arb{H}\aftergroup\egroup\right) \ar[r] & 0 \\
& & \mathfrak{X}\mathopen{}\mathclose\bgroup\left(G\aftergroup\egroup\right) \ar[r]_{\operatorname{res}} & \mathfrak{X}\mathopen{}\mathclose\bgroup\left(H\aftergroup\egroup\right) \ar[u]_{\cong}
}
\end{align*}
\end{prop}

\begin{proof}
The map $\mathfrak{X}\mathopen{}\mathclose\bgroup\left(\arb{C}\aftergroup\egroup\right) \oplus \mathbb{Z}^{\arb{\mathcal{D}}} \to
\Pic_{\arb{G}}\mathopen{}\mathclose\bgroup\left(\arb{G}/\arb{H}\aftergroup\egroup\right)$
sends $\chi \mapsto \mathcal{O}_{\arb{G}/\arb{H}}(\chi)$
for $\chi \in \mathfrak{X}(\arb{C})$
and $\eta_i \mapsto \mathcal{O}_{\arb{G}/\arb{H}}(\arb{D}_i)$
as in \cite[Proposition~4.1.1]{brcox}
where $\mathcal{O}_{\arb{G}/\arb{H}}(\arb{D}_i)$ is canonically
$\arb{G}$-linearized as in \cite[4.1]{brcox}.
The left-hand isomorphism is obvious,
$\operatorname{res}$ is the restriction map,
and the 
right-hand isomorphism sends $\chi \mapsto \mathscr{L}_{\arb{G}/\arb{H}}(-\chi)$ for
$\chi \in \mathfrak{X}(H)$ where
$\mathscr{L}_{\arb{G}/\arb{H}}(-\chi)$
is the standard construction (cf.~\cite[2.1]{local} or \cite[after Proposition~2.4]{ti}).
It is not difficult to see that the diagram commutes.
We denote by $\rho\mathopen{}\mathclose\bgroup\left(\arb{D}_i\aftergroup\egroup\right) \in \arb{\mathcal{N}}_\mathbb{Q}$ the vector corresponding to 
the restriction of the discrete valuation induced by $\arb{D}_i \in \arb{\mathcal{D}}$
and set $d_i(\mu) \coloneqq \mathopen{}\mathclose\bgroup\left(\rho\mathopen{}\mathclose\bgroup\left(\arb{D_i}\aftergroup\egroup\right)\aftergroup\egroup\right)(\mu) \in \mathbb{Z}$.
Then the map $\arb{\mathcal{M}} \to \mathfrak{X}\mathopen{}\mathclose\bgroup\left(\arb{C}\aftergroup\egroup\right) \oplus \mathbb{Z}^{\arb{\mathcal{D}}}$ sends
\begin{align*}
\mu \mapsto \sum_{i=1}^r d_i(\mu) \eta_i -
\mu|_{\arb{C}}\text{.}
\end{align*}
If $d_i(\mu) = 0$ for all $1 \le i \le r$ and $\mu$ is the $\arb{B}$-weight
of $f \in \mathbb{C}\mathopen{}\mathclose\bgroup\left(\arb{G}/\arb{H}\aftergroup\egroup\right)^{\mathopen{}\mathclose\bgroup\left(\arb{B}\aftergroup\egroup\right)}$, it follows that $\operatorname{div}(f) = 0$, so $f$ is
a unit in $\glob{\arb{G}/\arb{H}}$. This means we have $f \in 
\mathfrak{X}\mathopen{}\mathclose\bgroup\left(\arb{C}\aftergroup\egroup\right)$, so $\mu|_{\arb{C}} = 0$ if and only if $\mu = 0$.
Therefore the map is injective. Using the first statement of \cite[Proposition~4.1.1]{brcox},
which in fact does not require the assumption $\glob{\arb{G}/\arb{H}} = \mathbb{C}$,
we obtain the exactness of the top row.
\end{proof}

\begin{cor}
The spherical homogeneous space $G/H$ has trivial divisor class group.
\end{cor}

\begin{proof}
By a theorem of Popov, $\Cl(G/H) = \Pic(G/H)$ is trivial if and only if every character of $H$
is the restriction of a character of $G$ (cf.~\cite{Popov:pic}, \cite[Theorem~2.5]{ti}).
By Proposition~\ref{prop-ex-res}, $\operatorname{res}: \mathfrak{X}(G) \to \mathfrak{X}(H)$ is surjective.
\end{proof}

We will continue to use the notation from the previous sections
for the spherical homogeneous space $G/H$.
Where applicable, we use the same notation
for the spherical homogeneous space $\arb{G}/\arb{H}$,
but in boldface symbols. In particular,
$\arb{\mathcal{M}}$ is the weight lattice of $\arb{B}$-eigenvectors
in the function field $\mathbb{C}\mathopen{}\mathclose\bgroup\left(\arb{G}/\arb{H}\aftergroup\egroup\right)$, $\arb{\mathcal{N}} = \Hom\mathopen{}\mathclose\bgroup\left(\arb{\mathcal{M}}, \mathbb{Z}\aftergroup\egroup\right)$
the dual lattice, $\arb{\mathcal{N}}_\mathbb{Q} = \arb{\mathcal{N}} \otimes_\mathbb{Z} \mathbb{Q}$,
and $\arb{\mathcal{V}}$ is the valuation cone of $\arb{G}/\arb{H}$.

We fix a convenient choice for
the prime elements $f_1, \ldots, f_r \in \glob{G/H}$.
We have a natural inclusion $\epsilon: \mathbb{Z}^{\arb{\mathcal{D}}} \hookrightarrow \mathbb{C}[G]^*$, $\chi \mapsto \epsilon^\chi$. Then $f_i \coloneqq
\arb{f}_i\epsilon^{-\eta_i} \in \mathbb{C}[G]$ is $H$-invariant under the action from the right, and
$f_i$ is a prime element in $\glob{G/H}$ with $\mathbb{V}(f_i) = D_i$.
The torus $(\mathbb{C}^*)^{\arb{\mathcal{D}}}$ acts on $f_i \in \glob{G/H}$ with weight $\eta_i$.

Recall the decomposition $\mathcal{M} = \mathcal{M}_V \oplus \mathcal{M}_T$. We have a corresponding
decomposition of the dual lattice $\mathcal{N} = \mathcal{N}_V \oplus \mathcal{N}_T$.

We consider a spherical embedding $\arb{G}/\arb{H} \hookrightarrow \arb{Y}$
which contains only non-open $\arb{G}$-orbits of codimension one,
given by a fan $\arb{\Sigma}$ in $\arb{\mathcal{N}}_\mathbb{Q}$,
and assume $\Gamma\mathopen{}\mathclose\bgroup\left(\arb{Y}, \mathcal{O}_{\arb{Y}}^*\aftergroup\egroup\right) = \mathbb{C}^*$. We denote
by $\arb{Y}_1, \ldots, \arb{Y}_n$ the $\arb{G}$-invariant prime
divisors in $\arb{Y}$.

The next two results will allow us to obtain
a fan $\Sigma$ in $\mathcal{N}_\mathbb{Q}$ with associated spherical
embedding $G/H \hookrightarrow Y$ from the fan $\arb{\Sigma}$ in $\arb{\mathcal{N}}_\mathbb{Q}$.
We will then construct the ring $\mathcal{R}(Y)$ exactly as in Section~\ref{two}. As
we may have $\Gamma(Y, \mathcal{O}_Y^*) \ne \mathbb{C}^*$, the ring $\mathcal{R}(Y)$ may not be the Cox ring of $Y$.
This ring will, however, be the Cox ring $\mathcal{R}\mathopen{}\mathclose\bgroup\left(\arb{Y}\aftergroup\egroup\right)$ of $\arb{Y}$.

The quotient map $\arb{\pi}: G/H \to \arb{G}/\arb{H}$ induces an inclusion
$\arb{\pi}^*: \arb{\mathcal{M}} \hookrightarrow \mathcal{M}$,
which is the restriction of the natural inclusion
$\arb{\pi}^+: \mathfrak{X}\mathopen{}\mathclose\bgroup\left(\arb{B}\aftergroup\egroup\right) \hookrightarrow \mathfrak{X}(B)$.
We also obtain the surjective dual map
$\arb{\pi}_*: \mathcal{N}_\mathbb{Q} \to \arb{\mathcal{N}}_\mathbb{Q}$.

\begin{prop}
\label{prop-valcone}
We have
\begin{align*}
\mathcal{V} = \arb{\pi}_*^{-1}\mathopen{}\mathclose\bgroup\left(\arb{\mathcal{V}}\aftergroup\egroup\right)\text{.}
\end{align*}
\end{prop}
\begin{proof}
Let $v \in \mathcal{N}_\mathbb{Q}$ with $\arb{\pi}_*(v) = 0$.
Interpreting $v$ as map $v: \mathcal{M} \to \mathbb{Q}$, this means $v \circ \arb{\pi}^* = 0$.
Therefore there exists an extension $v^+: \mathfrak{X}(B) \to \mathbb{Q}$ of $v$ with $v^+ \circ \arb{\pi}^+ = 0$.
By \cite[Corollary~5.3]{knopsph}, we obtain $v \in \mathcal{V}$, therefore $\arb{\pi}_*^{-1}(0) \subseteq
\mathcal{V}$. Finally, we use \cite[Corollary~1.5]{knopsph}.
\end{proof}

As the $B$-weights of the kernel of the restriction map $\operatorname{res}: \mathfrak{X}(G) \to \mathfrak{X}(H)$ are exactly
the lattice $\mathcal{M}_T$, Proposition~\ref{prop-ex-res} yields an isomorphism
$\gamma: \arb{\mathcal{M}} \to \mathcal{M}_T$.

\begin{prop}
\label{prop-composedmap}
The restricted map
\begin{align*}
\arb{\pi}_*|_{\mathopen{}\mathclose\bgroup\left(\mathcal{N}_T\aftergroup\egroup\right)_\mathbb{Q}}: \mathopen{}\mathclose\bgroup\left(\mathcal{N}_T\aftergroup\egroup\right)_\mathbb{Q} \to \arb{\mathcal{N}}_\mathbb{Q}
\end{align*}
is dual to the isomorphism $\gamma: \arb{\mathcal{M}} \to \mathcal{M}_T$. In particular,
it is itself an isomorphism.
\end{prop}
\begin{proof}
We have to show that for each $v \in \mathcal{N}_\mathbb{Q}$ with $v|_{\mathcal{M}_V} = 0$
and each $\mu \in \arb{\mathcal{M}}$
we have $v(\gamma(\mu)) = v(\arb{\pi}^*(\mu))$. It therefore
suffices to show that $\gamma(\mu) - \arb{\pi}^*(\mu) \in \mathcal{M}_V$ for
each  $\mu \in \arb{\mathcal{M}}$.

Let $\mu \in \arb{\mathcal{M}}$ be the $\arb{B}$-weight of the
$\arb{B}$-eigenvector
$f \in \mathbb{C}\mathopen{}\mathclose\bgroup\left(\arb{G}/\arb{H}\aftergroup\egroup\right) \subseteq \mathbb{C}(\arb{G})$.
Using the notation from the proof of Proposition~\ref{prop-ex-res}, we
necessarily have
\begin{align*}
f = c\prod_{i=1}^r \arb{f}_i^{d_i(\mu)}\text{,}
\end{align*}
where $c \in \mathbb{C}\mathopen{}\mathclose\bgroup\left[\arb{G}\aftergroup\egroup\right]^*$ has left $\arb{B}$-weight $\mu|_{\arb{C}}$. Therefore we obtain
\begin{align*}
\gamma(\mu) = -\sum_{i=1}^r d_i(\mu)\eta_i + \mu|_{\arb{C}} && \text{ and } &&
\arb{\pi}^*(\mu) = \mu|_{\arb{C}} + \sum_{i=1}^r d_i(\mu)\omega_i
\end{align*}
where $\omega_i$ is the left
$\arb{B}$-weight of $\arb{f}_i$, hence $\gamma(\mu) - \arb{\pi}^*(\mu) \in \mathcal{M}_V$.
\end{proof}

We obtain the fan $\Sigma$ in $\mathcal{N}_\mathbb{Q}$
(with associated spherical embedding $G/H \hookrightarrow Y$)
as preimage under $\arb{\pi}_*|_{\mathopen{}\mathclose\bgroup\left(\mathcal{N}_T\aftergroup\egroup\right)_\mathbb{Q}}$ of the fan $\arb{\Sigma}$ in $\arb{\mathcal{N}}_\mathbb{Q}$.

\begin{remark}
There is a good geometric quotient of the whole toric variety $X$
by the action of $(\mathbb{C}^*)^{\arb{\mathcal{D}}}$. This means that $\arb{\pi}$ can be
extended to a quotient $\arb{\pi}: X \to \arb{X}$.
We obtain the following natural commutative diagram.
\begin{align*}
\xymatrix{
\mathopen{}\mathclose\bgroup\left(\mathcal{N}_T\aftergroup\egroup\right)_\mathbb{Q} \ar@{^{(}->}[r] & \mathcal{N}_\mathbb{Q} \ar[r]^{\arb{\pi}_*} \ar@{^{(}->}[d] & \arb{\mathcal{N}}_\mathbb{Q} \ar@{^{(}->}[d] \\
& N_\mathbb{Q} \ar[r]^{\arb{\pi}_*} & \arb{N}_\mathbb{Q}
}
\end{align*}
Similarly to Section~\ref{two}, we obtain $\arb{Y}$ as
closure of $\arb{G}/\arb{H}$ inside $\arb{X}$, and the embedding $\arb{Y} \hookrightarrow \arb{X}$ is neat.
\end{remark}

\begin{theorem}
\label{th-cox-gen}
We have
\begin{align*}
\mathcal{R}\mathopen{}\mathclose\bgroup\left(\arb{Y}\aftergroup\egroup\right) \cong S(V)[W_1, \ldots, W_n]\mathopen{}\mathclose\bgroup\left/\middle(f^h; f \in \mathfrak{p}\aftergroup\egroup\right)\text{,}
\end{align*}
with $\Cl\mathopen{}\mathclose\bgroup\left(\arb{Y}\aftergroup\egroup\right)$-grading given by $\deg(S_{ij}) = \mathopen{}\mathclose\bgroup\left[\arb{D}_i\aftergroup\egroup\right]$ and $\deg(W_l) = \mathopen{}\mathclose\bgroup\left[\arb{Y}_l\aftergroup\egroup\right]$.
\end{theorem}
\begin{proof}
As in Section~\ref{two}, we construct the good quotient
$p: \widehat{X} \to X$, and
we have the canonical quotient construction
$\pi: \widetilde{X} \to \arb{X}$
as well as a canonical toric closed embedding
$\phi: \widetilde{X} \to \widehat{X}$
such that $\pi = \arb{\pi} \circ p \circ \phi$. The result
now follows as in Section~\ref{two}.
\end{proof}

\begin{theorem}
If $\arb{H}$ is connected, $\mathcal{R}\mathopen{}\mathclose\bgroup\left(\arb{Y}\aftergroup\egroup\right)$ is a factorial ring.
\end{theorem}
\begin{proof}
We embed $Y$ into another toric variety $X'$ using primes $f'_i \in \glob{G/H}$ 
satisfying the requirements of the proof of Theorem \ref{thm-fact-fact} instead of the primes $f_i$.
The $f'_i$ can be chosen in such a way that for each $1 \le i \le r$ there is a unit $c_i$
such that $f'_i = c_if_i$ and $c_i$ is the product of elements of $\{g_k\}_{k=1}^m$.
As a basis for the $G$-module $\langle G\cdot f'_i\rangle$ we then choose
$\{c_if_{ij}\}_{j=1}^{s_i}$. In this case, there is a toric isomorphism
$X' \cong X$ which fixes $G/H$ and therefore $Y$ as well.
We obtain the following commutative diagram.
\begin{align*}
\xymatrix{
& \widehat{X}' 
\ar[r]^{p'} & X' \ar[d]^\cong  \\
\widetilde{X} \ar[r]_{\phi} \ar[ru]^{\phi'} & \widehat{X} \ar[r]_p &
X \ar[r]_{\arb{\pi}} & \arb{X}
}
\end{align*}
The factoriality of $\mathcal{R}\mathopen{}\mathclose\bgroup\left(\arb{Y}\aftergroup\egroup\right)$ now follows as in Theorem \ref{thm-fact-fact}.
\end{proof}

The following Theorem~\ref{th-cox-hor} follows directly
from \cite[Theorem~4.3.2]{brcox}. In order to be self-contained,
we give another proof.

\begin{theorem}
\label{th-cox-hor}
If $\arb{G}/\arb{H}$ is horospherical, we have
\begin{align*}
\mathcal{R}\mathopen{}\mathclose\bgroup\left(\arb{Y}\aftergroup\egroup\right) \cong \mathcal{R}\mathopen{}\mathclose\bgroup\left(\arb{G}/\arb{P}\aftergroup\egroup\right)[W_1, \ldots W_n]\text{,}
\end{align*}
where $\arb{P} \coloneqq N_{\arb{G}}\mathopen{}\mathclose\bgroup\left(\arb{H}\aftergroup\egroup\right)$.
\end{theorem}

\begin{proof}
As $\arb{G}/\arb{H}$ being horospherical implies
that $G/H$ is horospherical as well, the ideal $\mathfrak{p}$ is $\mathcal{M}$-graded
by Proposition \ref{prop-grading}. This means,
for each $f \in \mathfrak{p}$ and $\mu \in \mathcal{M}$ we
have $f^{(\mu)} \in \mathfrak{p}$. As $\mathopen{}\mathclose\bgroup\left(f^{(\mu)}\aftergroup\egroup\right)^\alpha = f^{(\mu)}$,
the ideal $(f^h; f \in \mathfrak{p})$ can be generated by elements which do not
contain any of the variables $W_l$. It follows that
$\mathcal{R}\mathopen{}\mathclose\bgroup\left(\arb{Y}\aftergroup\egroup\right) \cong R[W_1, \ldots, W_n]$ for some ring $R$
which depends only on the homogeneous space.

It only remains to show that $R \cong \mathcal{R}\mathopen{}\mathclose\bgroup\left(\arb{G}/\arb{P}\aftergroup\egroup\right)$.
We have canonical maps
\begin{align*}
\xymatrix{
\arb{G} \ar[r] & \arb{G}/\arb{H} \ar[r]^-{\cong} & 
\mathopen{}\mathclose\bgroup\left(\arb{G} \times \arb{T}\aftergroup\egroup\right)/\arb{P} \ar[r] & \arb{G}/\arb{P}\text{,}
}
\end{align*}
where $\arb{T} \coloneqq \arb{P}/\arb{H}$ is a torus, 
$\arb{P}$ acts on $\arb{T}$ via $\arb{P} \to \arb{T}$, and $\arb{P}$ acts
on $\arb{G} \times \arb{T}$ via $p \cdot (g, t) \coloneqq (gp^{-1}, pt)$.
The last map has fibers isomorphic to $\arb{T}$ and is a trivial fibration
over the open orbit of any Borel subgroup of $\arb{G}$ (cf.~\cite[after Theorem~2.2]{bm}).
The pullbacks of the $\arb{B}$-invariant prime divisors in $\arb{G}/\arb{P}$
are exactly the $\arb{B}$-invariant prime divisors in $\arb{G}/\arb{H}$.
In particular, $\arb{P}$ acts from the right on each $\arb{f}_i \in \mathbb{C}\mathopen{}\mathclose\bgroup\left[\arb{G}\aftergroup\egroup\right]$
with an extension of the character $\arb{\chi}_i \in \mathfrak{X}\mathopen{}\mathclose\bgroup\left(\arb{H}\aftergroup\egroup\right)$ which we
will also call $\arb{\chi}_i \in \mathfrak{X}\mathopen{}\mathclose\bgroup\left(\arb{P}\aftergroup\egroup\right)$. We define
\begin{align*}
P &\coloneqq \mathopen{}\mathclose\bgroup\left\{\mathopen{}\mathclose\bgroup\left(p, \arb{\chi}_1(p), \ldots, \arb{\chi}_r(p)\aftergroup\egroup\right); p \in \arb{P}\aftergroup\egroup\right\} \subseteq G\text{.}
\end{align*}
It is not difficult to see that $G/H$ is isomorphic to $\mathopen{}\mathclose\bgroup\left(G \times \arb{T}\aftergroup\egroup\right)/P$
where $P \cong \arb{P}$ acts on $G \times \arb{T}$ via $p \cdot (g, t) \coloneqq (gp^{-1}, pt)$.
As all characters of $P$ can be extended to $G$ (cf.~Proposition \ref{prop-ex-res}), we
obtain $G/H \cong G/P \times \arb{T}$, and the result follows.
\end{proof}

\section{Examples}
\label{four}

\begin{example}
Consider $G \coloneqq \SL(3)$ and $H \coloneqq U$, the set of unipotent
upper triangular matrices. 
Let $B \subseteq G$ be the Borel subgroup
of upper triangular matrices, and let $G$ act linearly on $\mathbb{C}^3 \times \mathbb{C}^3$ by acting
naturally on the first factor and with the contragredient action on the second
factor. We denote the coordinates of the first factor by $S_{11}, S_{12}, S_{13}$
and the coordinates of the second factor by $S_{21}, S_{22}, S_{23}$. Then the point
$((1,0,0), (0,0,1))$ has isotropy group $U$, and its orbit is
$\mathbb{V}(S_{11}S_{21}+S_{12}S_{22}+S_{13}S_{23})$. The homogeneous space
$G/H$ is horospherical with $B$-invariant prime divisors $D_1 \coloneqq \mathbb{V}(S_{13})$ and $D_2 \coloneqq \mathbb{V}(S_{21})$
and has trivial divisor class group. Consider the embedding $Y$ corresponding
to a fan containing exactly $n$ one-dimensional cones.
By Theorem~\ref{th-cox-hor} and setting $P\coloneqq N_G(H)$, we obtain
\begin{align*}
\mathcal{R}(G/P) &\cong \mathbb{C}[S_{11}, S_{12}, S_{13}, S_{21}, S_{22}, S_{23}] \mathopen{}\mathclose\bgroup\left/ \middle(S_{11}S_{21}+S_{12}S_{22}+S_{13}S_{23}\aftergroup\egroup\right) \\
\mathcal{R}(Y) &\cong \mathcal{R}(G/P)[W_1, \ldots, W_n]\text{.}
\end{align*}
\end{example}

\begin{example}
\label{ex:comp}
Consider $G \coloneqq \SL(d)$ for $d \ge 3$ and $H \coloneqq \SL(d-1)$ embedded
as the lower-right entries of $\SL(d)$. The case $d=3$ has
been studied in \cite{pauer1} and \cite{pauersu}. Let $B \subseteq G$ be the Borel subgroup
of upper triangular matrices, and let $G$ act linearly on $\mathbb{C}^d \times \mathbb{C}^d$ by acting
naturally on the first factor and with the contragredient action on the second
factor. We denote the coordinates of the first factor by $S_{11}, \ldots, S_{1d}$
and the coordinates of the second factor by $S_{21}, \ldots, S_{2d}$. Then the point
$((1,0,\ldots,0), (1,0,\ldots,0))$ has isotropy group $\SL(d-1)$, and its orbit is
$\mathbb{V}\big(\sum_{j=1}^d S_{1j}S_{2j} - 1\big)$. The homogeneous space
$G/H$ is spherical with $B$-invariant prime divisors $D_1 \coloneqq \mathbb{V}(S_{1d})$ and $D_2 \coloneqq \mathbb{V}(S_{21})$
as well as affine with factorial coordinate ring.
By Theorem~\ref{prop-trop}, we obtain $\mathcal{V} = \{v_1^* + v_2^* \le 0\}$.
Consider the embedding $Y$ corresponding to the fan containing
the one-dimensional cones having primitive lattice generators
$(p_1, q_1), \ldots, (p_n, q_n) \in \mathcal{N}$ with respect
to the basis $\{v_1, v_2\}$.
The following picture illustrates $\mathcal{N}_\mathbb{Q}$.
\begin{align*}
\begin{tikzpicture}
     \begin{scope}[help lines]
     \draw (-1.5,0) -- (1.5,0);
     \draw (0,-1.5) -- (0,1.5);
     \draw (-1.5, 1.5) -- (1.5, -1.5);
     \end{scope}
     \fill[opacity=0.1] (-1.5, 1.5) -- (1.5, -1.5) -- (-1.5, -1.5) -- cycle;
     \draw (1,-1pt) -- (1,1pt) node[anchor=south west]{\tiny{$v_1 = \rho(D_1)$}};
     \draw (-1pt,1) -- (1pt,1) node[anchor=south west]{\tiny{$v_2 = \rho(D_2)$}};
     \fill (1, 0) circle (2pt) ;
     \fill (0, 1) circle (2pt) ;
     \path (-0.75, -0.75) node {{$\mathcal{V}$}};
\end{tikzpicture}
\end{align*}
By Theorem~\ref{th-cox}, we obtain 
\begin{align*}
\mathcal{R}(Y) \cong \mathbb{C}[S_{1j}, S_{2j}, W_1, \ldots, W_n]_{j=1}^d \mathopen{}\mathclose\bgroup\left/ \middle(\sum_{j=1}^d S_{1j}S_{2j} - W_1^{-p_1-q_1}\cdots W_n^{-p_n-q_n}\aftergroup\egroup\right) \text{.}
\end{align*}
\end{example}

\begin{example}
Consider $\arb{G} \coloneqq \SL(2)$ and $\arb{H} \coloneqq T$,
the diagonal torus. Let $\arb{B} \subseteq \arb{G}$ be the
Borel subgroup of upper triangular matrices. Then, considering 
 the orbit of $([1:0], [0:1])$ in $\mathbb{P}^1 \times \mathbb{P}^1$, we obtain
$\arb{G}/\arb{H} \cong \mathbb{P}^1 \times \mathbb{P}^1 \setminus \operatorname{diag}\mathopen{}\mathclose\bgroup\left(\mathbb{P}^1\aftergroup\egroup\right)$,
and there are two $\arb{B}$-invariant prime divisors. From
\begin{align*}
\mathbb{C}\mathopen{}\mathclose\bgroup\left[\arb{G}\aftergroup\egroup\right] = \mathbb{C}[M_{11}, M_{12}, M_{21}, M_{22}]/\mathopen{}\mathclose\bgroup\left(M_{11}M_{22} - M_{12}M_{21} - 1\aftergroup\egroup\right)\text{,}
\end{align*}
we obtain $\arb{f}_1 = M_{21}$ and $\arb{f}_2 = M_{22}$. We can now construct $G = \arb{G} \times
\mathopen{}\mathclose\bgroup\left(\mathbb{C}^*\aftergroup\egroup\right)^{\arb{\mathcal{D}}}$
and $H$. It is not difficult to see that $G/H$ is the
orbit of the point $((1,0), (0,1), 1)$ in $\mathbb{C}^2 \times \mathbb{C}^2 \times \mathbb{C}^*$,
where $\arb{G}$ acts naturally on both factors $\mathbb{C}^2$ and trivially on $\mathbb{C}^*$ while
$\mathopen{}\mathclose\bgroup\left(\mathbb{C}^*\aftergroup\egroup\right)^{\arb{\mathcal{D}}}$
acts with the weights $-\eta_1$ and $-\eta_2$ respectively on the factors $\mathbb{C}^2$
as well as with $-\eta_1 -\eta_2$ on $\mathbb{C}^*$.
Denoting by $S_{11}$, $S_{12}$ and $S_{21}$, $S_{22}$ the coordinates of the factors $\mathbb{C}^2$
and by $T_1$ the coordinate of the factor $\mathbb{C}^*$, we obtain
$G/H = \mathbb{V}\mathopen{}\mathclose\bgroup\left(S_{11}S_{22} - S_{12}S_{21} - T_1\aftergroup\egroup\right)$.
By Theorem~\ref{prop-trop}, we obtain $\mathcal{V} = \{v_1^* + v_2^* \le w_1^*\}$. Using
Proposition~\ref{prop-composedmap} and Proposition~\ref{prop-ex-res},
we see that $\arb{\pi}_*(w_1)$ is a basis of $\arb{\mathcal{N}}_\mathbb{Q}$ and that
$\arb{\pi}_*(v_1) = \arb{\pi}_*(v_2) = -\arb{\pi}_*(w_1)$.
By Proposition~\ref{prop-valcone}, $\arb{\mathcal{V}} = \{\arb{\pi}_*(w_1)^* \ge 0\}$ follows.
The following picture illustrates $\arb{\mathcal{N}}_\mathbb{Q}$.
\begin{align*}
\begin{tikzpicture}
     \begin{scope}[help lines]
     \draw (-2,0) -- (2,0);
     \draw (0, -0.1) -- (0, 0.1);
     \end{scope}
     \fill[opacity=0.1] (2, -0.05) -- (0, -0.05) -- (0, 0.05) -- (2, 0.05) -- cycle;
     \draw (1,-1pt) -- (1,1pt) node[anchor=south]{\tiny{$\arb{\pi}_*(w_1)$}};
     \fill (-1, 0) circle (2pt) node[anchor=north]{\tiny{$\rho(\arb{D}_2)$}} node[anchor=south]{\tiny{$\rho(\arb{D}_1)$}} ;
     \path (1, 0) node[anchor=north] {\tiny{$\arb{\mathcal{V}}$}};
\end{tikzpicture}
\end{align*}
By Theorem~\ref{th-cox-gen}, we have
\begin{align*}
\mathcal{R}\mathopen{}\mathclose\bgroup\left(\arb{G}/\arb{H}\aftergroup\egroup\right) \cong \mathbb{C}[S_{11},S_{12},S_{21},S_{22}]/\mathopen{}\mathclose\bgroup\left(S_{11}S_{22} - S_{12}S_{21}-1\aftergroup\egroup\right)\text{.}
\end{align*}

The only nontrivial embedding of $\arb{G}/\arb{H}$ is the embedding
$\arb{Y} \coloneqq \mathbb{P}^1 \times \mathbb{P}^1$ given by the cone $\mathbb{Q}_{\ge 0}\arb{\pi}_*(w_1)$.
In this case, Theorem~\ref{th-cox-gen} yields
\begin{align*}
\mathcal{R}\mathopen{}\mathclose\bgroup\left(\arb{Y}\aftergroup\egroup\right) \cong \mathbb{C}[S_{11},S_{12},S_{22},S_{21},W_1]/\mathopen{}\mathclose\bgroup\left(S_{11}S_{22} - S_{12}S_{21}-W_1\aftergroup\egroup\right)\text{,}
\end{align*}
which is isomorphic to the polynomial ring in four variables.
\end{example}

\begin{example}[{cf.~\cite[Example~4.2]{cfact}}]
Consider $\arb{G} \coloneqq \SL(2)$ and $\arb{H} \coloneqq N_{\arb{G}}\mathopen{}\mathclose\bgroup\left(T\aftergroup\egroup\right)$
where $T$ is the diagonal torus.
Let $\arb{B} \subseteq \arb{G}$ be the Borel subgroup of upper triangular matrices.
The group $\arb{H}$ consists of two connected components, the identity
component
$\arb{H}^\circ = \big\{\big(\begin{smallmatrix}\lambda & 0 \\ 0 & \lambda^{-1}\end{smallmatrix}\big); \lambda \in \mathbb{C}^*\big\}$ and a second component
$\arb{H}^\dagger = \big\{\big(\begin{smallmatrix} 0 & \lambda & \\ -\lambda^{-1} & 0\end{smallmatrix}\big); \lambda \in \mathbb{C}^*\big\}$. There exists a character $\arb{\chi} \in \mathfrak{X}\mathopen{}\mathclose\bgroup\left(\arb{H}\aftergroup\egroup\right)$
with $\arb{\chi}|_{\arb{H}^\circ} = 1$ and $\arb{\chi}|_{\arb{H}^\dagger} = -1$.
Consider the complex vector space $\Sym(2\times 2) = 
\big\{\big(\begin{smallmatrix}s_{12} & s_{13} \\ s_{13} & s_{11}\end{smallmatrix}\big); s_{ij} \in \mathbb{C}^*\big\}$
of symmetrical 2 by 2 matrices. Let $\arb{G} \times \mathbb{C}^*$
act on $\Sym(2\times 2) \times \mathbb{C}^*$ via $(g, t)\cdot (x, y) \coloneqq (t^{-1}gxg^T, t^{-2}y)$. The isotropy
group of the point $\big(\big(\begin{smallmatrix}0 & 1 \\ 1 & 0\end{smallmatrix}\big), 1\big)$ is $H' \coloneqq \mathopen{}\mathclose\bgroup\left\{(h, \arb{\chi}(h)), h \in \arb{H}\aftergroup\egroup\right\}$ and its orbit is the
closed subset $\mathbb{V}(S_{11}S_{12} - S_{13}^2 - T_1)$. We observe that $\mathopen{}\mathclose\bgroup\left(\arb{G} \times \mathbb{C}^*\aftergroup\egroup\right)/H'$ is
a spherical homogeneous space with one $\arb{B} \times \mathbb{C}^*$-invariant
prime divisor $D_1 \coloneqq \mathbb{V}(S_{11})$ and that it
coincides with $G/H$ as defined in Section~\ref{three}. By Theorem~\ref{prop-trop}, we
obtain $\mathcal{V} = \{2v_1^* \le w_1^*\}$. Using
Proposition~\ref{prop-composedmap} and Proposition~\ref{prop-ex-res},
we see that $\arb{\pi}_*(w_1)$ is a basis of $\arb{\mathcal{N}}_\mathbb{Q}$ and that
$\arb{\pi}_*(v_1) = -2\arb{\pi}_*(w_1)$.
By Proposition~\ref{prop-valcone}, $\arb{\mathcal{V}} = \{\arb{\pi}_*(w_1)^* \ge 0\}$ follows.
The following picture illustrates $\arb{\mathcal{N}}_\mathbb{Q}$.
\begin{align*}
\begin{tikzpicture}
     \begin{scope}[help lines]
     \draw (-2.5,0) -- (2.5,0);
     \draw (0, -0.1) -- (0, 0.1);
     \end{scope}
     \fill[opacity=0.1] (2.5, -1pt) -- (0, -1pt) -- (0, 1pt) -- (2.5, 1pt) -- cycle;
     \draw (2,-1pt) -- (2,1pt) ;
     \draw (-1,-1pt) -- (-1,1pt) ;
     \draw (1,-1pt) -- (1,1pt) node[anchor=south]{\tiny{$\arb{\pi}_*(w_1)$}};
     \fill (-2, 0) circle (2pt) node[anchor=south]{\tiny{$\rho(\arb{D_1})$}} ;
     \path (1.25, 0) node[anchor=north] {\tiny{$\arb{\mathcal{V}}$}};
\end{tikzpicture}
\end{align*}
Consider the trivial embedding $\arb{Y} \coloneqq \arb{G}/\arb{H}$.
From a geometric viewpoint, this is the complement
of a conic in $\mathbb{P}^2$.
We have $\Cl\mathopen{}\mathclose\bgroup\left(\arb{Y}\aftergroup\egroup\right) \cong \mathbb{Z}/2\mathbb{Z}$ with generator 
$\mathopen{}\mathclose\bgroup\left[\arb{D}_1\aftergroup\egroup\right] \in \Cl\mathopen{}\mathclose\bgroup\left(\arb{Y}\aftergroup\egroup\right)$.
By Theorem~\ref{th-cox-gen}, we obtain
\begin{align*}
\mathcal{R}\mathopen{}\mathclose\bgroup\left(\arb{Y}\aftergroup\egroup\right) \cong \mathbb{C}[S_{11}, S_{12}, S_{13}]/(S_{11}S_{12} - S_{13}^2 - 1)\text{,}
\end{align*}
which is a non-factorial ring.
\end{example}

\section{Comparison with the approach of Brion}
\label{five}

When $Y$ is a spherical $G$-variety satisfying $\glob{Y} = \mathbb{C}$, Brion defines the
\em equivariant Cox ring \em of $Y$, which is graded by 
the group $\Cl_{G}(Y)$ of isomorphism classes of $G$-linearized
divisorial sheaves on $Y$, as
\begin{align*}
\mathcal{R}_{G}(Y) \coloneqq \bigoplus_{[\mathcal{F}] \in \Cl_{G}(Y)} \Gamma(Y, \mathcal{F})\text{,}
\end{align*}
where again some accuracy is required in order to define a multiplication law (cf.~\cite[4.2]{brcox}).
We have $\mathcal{R}_G(Y) \cong \mathcal{R}(Y) \otimes_\mathbb{C} \mathbb{C}[C]$, where the characters of the torus
$C$ correspond to the various choices of linearizations.

In order to obtain a description of $\mathcal{R}_G(Y)$, Brion first computes the Cox ring
of the associated wonderful variety $Y'$.
According to \cite[6.1]{Luna:typea}, the wonderful variety $Y'$ is obtained as follows.
The equivariant automorphism group of $G/H$ can be identified with $N_G(H)/H$, hence
$N_G(H)$ acts on the set $\mathcal{D}$ of $B$-invariant prime divisors in $G/H$.
The subgroup of $N_G(H)$ which stabilizes $\mathcal{D}$ is called the \em spherical closure \em of $H$. 
It can be written as $H' \times C$ where $H' \subseteq G' \coloneqq G^{ss}$.
Then $G'/H'$ is a spherical homogeneous space admitting
a wonderful completion $G'/H' \hookrightarrow Y'$.

We consider the Borel subgroup $B' \coloneqq B \cap G'$ as well as the maximal torus $T' \coloneqq T \cap G'$,
define $\mathfrak{X} \coloneqq \mathfrak{X}(B') = \mathfrak{X}(T')$, and denote by $\mathfrak{X}^+ \subseteq \mathfrak{X}$ the subset of dominant weights.
There is a natural inclusion $\epsilon: \mathfrak{X} \hookrightarrow \mathbb{C}[T']^*$, $\chi \mapsto \epsilon^\chi$.
The set of $B'$-invariant prime divisors in $G'/H'$ can be identified with $\mathcal{D} = \{D_1, \ldots, D_r\}$.
We denote by $\omega_i$ the left $B'$-weight of the equation $f'_i$ of the
pullback of $D_i$ under $G' \to G'/H'$. There is a natural bijection
between the set $\{\gamma_1, \ldots, \gamma_s\}$ of spherical roots of $G'/H'$
and the set  $\{Y'_1, \ldots, Y'_s\}$ of $G'$-invariant
prime divisors in $Y'$.

The Cox ring $\mathcal{R}(Y')$ is realized as a subring of
\begin{align*}
\mathbb{C}[G' \times T'] = \mathbb{C}[G'] \otimes_{\mathbb{C}} \mathbb{C}[T'] = \bigoplus_{\eta \in \mathfrak{X}} \mathbb{C}[G']\otimes_\mathbb{C} \mathbb{C}\epsilon^\eta\text{.}
\end{align*}
We will require the following notation: when $V$ is a $G'$-module and $\chi \in \mathfrak{X}^+$, we denote by $V_{(\chi)}$
the corresponding isotypic component of $V$, i.e.~the sum of all
irreducible $G'$-submodules of highest weight $\chi$.
We denote by $s_l \in \Gamma(Y', \mathcal{O}_{Y'}(Y'_l))$ and $s_{D_i} \in \Gamma(Y', \mathcal{O}_{Y'}(D_i))$
the respective canonical sections.
Finally, we denote by $K'$ the intersection of the kernels of all characters of $H'$.
Note that $G'/K'$ is always quasiaffine, but
may fail to have trivial divisor class group when
$H'$ is not connected.


\begin{theorem}[cf.~{\cite[Theorem~3.2.3]{brcox}}]
\label{thm:brionw}
We define the following partial order on $\mathfrak{X}$: we write $\chi \preceq \eta$ if $\eta - \chi$
is a linear combination of spherical roots of $G'/H'$ with nonnegative coefficients.
Then we have
\begin{align*}
\mathcal{R}(Y') \cong \bigoplus_{\substack{(\chi, \eta) \in \mathfrak{X}^+ \times \mathfrak{X}\\ \chi \preceq \eta}}
\glob{G'/K'}_{(\chi)} \otimes_\mathbb{C} \mathbb{C}\epsilon^\eta\text{.}
\end{align*}
This isomorphism sends $s_l \mapsto 1 \otimes \epsilon^{\gamma_l}$ and $s_{D_i} \mapsto f'_i \otimes \epsilon^{\omega_i}$.
Furthermore, $\mathcal{R}(Y')$ is the Rees algebra associated to
the ascending filtration
\begin{align*}
\mathscr{F}^{\eta}(\glob{G'/K'}) \coloneqq \bigoplus_{\substack{\chi\in\mathfrak{X} \\ \chi \preceq \eta}} \glob{G'/K'}_{(\chi)}
\end{align*}
of $\glob{G'/K'}$ indexed by the partially ordered set $\mathfrak{X}$.
\end{theorem}

\begin{example}
\label{ex:w}
Let $G' \coloneqq \SL(d)$ for $d \ge 3$ act on $Y' \coloneqq \mathbb{P}^d \times \mathbb{P}^d$
by acting naturally on the first factor and with the
contragredient action on the second factor.
Then $Y'$ is a wonderful variety and the point $((1:0:\ldots:0), (1:0:\ldots:0)) \in Y'$ with
isotropy group $H' \coloneqq \GL(d-1)$ lies in the open $G'$-orbit.
We have
\begin{align*}
\mathcal{R}(Y') = \mathbb{C}[X_1, \ldots, X_d, Y_1, \ldots, Y_d]\text{.}
\end{align*}
There is exactly one $G$-invariant prime divisor
$D \coloneqq \mathbb{V}\mathopen{}\mathclose\bgroup\left(X_1Y_1 + \ldots + X_dY_d\aftergroup\egroup\right)$.
The intersection $K'$ of the kernels of all characters of $H'$ is $\SL(d-1)$ as in Example~\ref{ex:comp}.
In particular, we have
\begin{align*}
R \coloneqq \glob{G'/K'} = \mathbb{C}[X_1, \ldots, X_d, Y_1, \ldots, Y_d] \mathopen{}\mathclose\bgroup\left/ \middle(X_1Y_1 + \ldots + X_dY_d - 1\aftergroup\egroup\right) \text{.}
\end{align*}
With the notation from Example~\ref{ex:comp}, $\gamma_1 \coloneqq v_1^*+v_2^* \in \mathfrak{X}(B')$ is 
the unique spherical root of $G'/H'$. Theorem~\ref{thm:brionw} yields
an isomorphism
\begin{align*}
\mathcal{R}(Y') \cong \bigoplus_{i,j,k=0}^\infty R_{(iv_1^*+jv_2^*)} \otimes_\mathbb{C} \mathbb{C}\epsilon^{iv_1^*+jv_2^*+k\gamma_1}
\end{align*}
sending $X_i \mapsto X_i \otimes \epsilon^{v_1^*}$ and $Y_i \mapsto Y_i \otimes \epsilon^{v_2^*}$. In particular, it sends
\begin{align*}
X_1Y_1 + \ldots + X_dY_d \mapsto 1 \otimes \epsilon^{\gamma_1}\text{.}
\end{align*}
\end{example}

Now assume that the fan associated to the spherical embedding $G/H \hookrightarrow Y$ 
contains exactly the trivial cone $0$ and the one-dimensional cones with primitive lattice generators $u_1, \ldots, u_n \in \mathcal{N}$.
The natural
map $G/H \to G'/H'$ extends to a $G$-equivariant morphism
$\phi: Y \to Y'$,
which induces an equivariant homomorphism of graded algebras
\begin{align*}
\phi^*: \mathcal{R}(Y') = \mathcal{R}_{G'}(Y') \to \mathcal{R}_{G}(Y)\text{.}
\end{align*}
The invariant subrings
\begin{align*}
\mathcal{R}_{G}(Y)^{G'} &\cong \mathbb{C}[C][W_1, \ldots, W_n]\\
\mathcal{R}(Y')^{G'} &\cong \mathbb{C}[Z_1, \ldots, Z_s]
\end{align*}
are polynomial rings over $\mathbb{C}[C]$ and $\mathbb{C}$
where $W_1, \ldots, W_n$ and $Z_1, \ldots, Z_s$ are identified with the canonical sections
corresponding to the $G$-invariant prime divisors in $Y$ and $Y'$ respectively.

\begin{theorem}[{cf.~\cite[Theorem~4.3.2]{brcox}}]
The restricted map
\begin{align*}
\phi^*|_{\mathcal{R}(Y')^{G'}}: \mathcal{R}(Y')^{G'} \to \mathcal{R}_{G}(Y)^{G'}
\end{align*}
sends $Z_i \mapsto W_1^{-\langle u_1, \gamma_i \rangle} \cdots W_n^{-\langle u_n, \gamma_i \rangle}$, and the map
\begin{align*}
\mathcal{R}(Y') \otimes_{\mathcal{R}(Y')^{G'}} \mathcal{R}_{G}(Y)^{G'} \to \mathcal{R}_{G}(Y) 
\end{align*}
sending $s' \otimes s \mapsto \phi^*(s')s$ is an isomorphism.
\end{theorem}

\begin{remark}
As $\mathcal{R}(Y) \cong \mathcal{R}_G(Y)^C$ (cf.~\cite[proof of Theorem~4.3.2]{brcox}), it
immediately follows that
\begin{align*}
\mathcal{R}(Y) \cong \mathcal{R}(Y') \otimes_{\mathbb{C}[Z_1, \ldots, Z_s]} \mathbb{C}[W_1, \ldots, W_n]\text{.}
\end{align*}
Furthermore, if $Y$ is horospherical and $P \coloneqq N_G(H)$, we have $Y' \cong G/P$
and $s = 0$, hence $\mathcal{R}(Y) \cong \mathcal{R}(G/P)[W_1, \ldots, W_n]$ (cf.~Theorem~\ref{th-cox-hor}).
\end{remark}

\begin{example}
Consider $G \coloneqq \SL(d)$ for $d \ge 3$ and $H \coloneqq \SL(d-1)$
as in Example~\ref{ex:comp}. Then $G' = G$, and the spherical closure of $H$
is $H' \coloneqq \GL(d-1)$ as in Example~\ref{ex:w}.
Consider the embedding $G/H \hookrightarrow Y$ corresponding to the fan containing
the one-dimensional cones having primitive lattice generators
$(p_1, q_1), \ldots, (p_n, q_n) \in \mathcal{N}$ with respect
to the basis $\{v_1, v_2\}$ as in Example~\ref{ex:comp}. There is
exactly one spherical root $\gamma_1 = v_1^* + v_2^*$.
As $G$ is semisimple, it follows that $\mathcal{R}_{G}(Y) = \mathcal{R}(Y)$.
We have
\begin{align*}
\mathcal{R}(Y') &= \mathbb{C}[X_1, \ldots, X_d, Y_1, \ldots, Y_d]\\
\mathcal{R}(Y')^{G} &= \mathbb{C}[Z_1] \\
\mathcal{R}(Y)^{G} &= \mathbb{C}[W_1, \ldots, W_n]\text{,}
\end{align*}
the inclusion
$\mathbb{C}[Z_1] \hookrightarrow \mathbb{C}[X_1, \ldots, X_d, Y_1, \ldots, Y_d]$ is given
by $Z_1 \mapsto X_1Y_1 + \ldots + X_dY_d$,
and $\phi^*: \mathbb{C}[Z_1] \to \mathbb{C}[W_1, \ldots, W_s]$ sends
$Z_1 \mapsto W_1^{-p_1-q_1}\cdots W_n^{-p_n-q_n}$.
From
\begin{align*}
\mathcal{R}(Y) \cong \mathbb{C}[X_1, \ldots, X_d, Y_1, \ldots, Y_d] \otimes_{\mathbb{C}[Z_1]} \mathbb{C}[W_1, \ldots, W_n]\text{,}
\end{align*}
we obtain
\begin{align*}
\mathcal{R}(Y) \cong \mathbb{C}[X_j,Y_j, W_1, \ldots, W_n]_{j=1}^d \mathopen{}\mathclose\bgroup\left/ \middle(\sum_{j=1}^d X_{j}Y_{j} - W_1^{-p_1-q_1}\cdots W_n^{-p_n-q_n}\aftergroup\egroup\right) \text{,}
\end{align*}
which is in agreement with Example~\ref{ex:comp}.
\end{example}

\section*{Acknowledgments}
I am highly grateful to my teacher, Victor Batyrev, for his guidance and support.
I would also like to thank J\"{u}rgen Hausen and Johannes Hofscheier for
some useful discussions. Finally, I thank the referee for carefully
reading the manuscript as well as for valuable comments and suggestions.

\bibliographystyle{amsalpha}
