2.1 Polarized toric varieties and polytopes

We refer to [Reference Cox, Little and Schenck2, chapters 2, 3, 6] and [22] for this section. Let $N$ be a rank $n$ lattice, $M:=\mathrm {Hom}_\mathbb {Z}(N,\,\mathbb {Z})$ be its dual with pairing $\langle \cdot,\,\cdot \rangle$. Then $N$ is the lattice of $1$-parameter subgroups of a $n$-dimensional complex torus

\[ T:=N\otimes_\mathbb{Z} \mathbb{C}^{*}= \mathrm{Hom}_\mathbb{Z}(M,\mathbb{C}^{*}). \]

We set $N_\mathbb {F}=N\otimes _\mathbb {Z} \mathbb {F}$ and $M_\mathbb {F}=M\otimes _\mathbb {Z} \mathbb {F}$ for $\mathbb {F}=\mathbb {Q}$ or $\mathbb {R}$.

Let $X=X(\Sigma )$ be an $n$-dimensional projective toric variety associated to a fan $\Sigma$, so that in particular $X$ is normal. Denote $\Sigma =\lbrace \sigma _i\, : i\in I \rbrace$, with $\sigma _i$ a strongly convex rational polyhedral cone in $N_\mathbb {R}$ for all $i\in I$. Denote also by $\Sigma (k)$ the set of $k$-dimensional cones in $\Sigma$. The variety $X$ is obtained by gluing affine charts $(U_\sigma )_{\sigma \in \Sigma }$, with

\[ U_\sigma=\mathrm{Spec}(\mathbb{C}[S_\sigma]), \]

and $\mathbb {C}[S_\sigma ]$ is the semi-group algebra of

\[ S_\sigma=\sigma^{{\vee}}\cap M =\lbrace m\in M: \langle m , n \rangle \geq 0 \text{ for all }n\in \sigma\rbrace . \]

There exists a bijective correspondence between cones $\sigma \in \Sigma$ and $T$-orbits $O(\sigma )$ in $X$. This satisfies, for $\sigma \in \Sigma$, $\dim O(\sigma )=n-\dim (\sigma )$. In particular, to any $\rho \in \Sigma (1)$, there corresponds a irreducible $T$-invariant Weil divisor $D_\rho$ given by

(2.1)\begin{equation} D_\rho=\overline{O(\rho)} \end{equation}

where the closure is in both classical and Zariski topologies.

As $X$ is associated to the fan $\Sigma$, there is a bijective correspondence between $T$-invariant ample divisors on $X$ and lattice polytopes $P\subset M_\mathbb {R}$ whose normal fan $\Sigma _P$ is equal to $\Sigma$ (see [Reference Cox, Little and Schenck2, theorem 6.2.1]). Let $D$ be a $T$-invariant Cartier divisor on $X$. Recall that it is equal to a linear combination of the form

\[ D =\sum_{\rho \in \Sigma{(1)}} a_\rho D_\rho. \]

For each $\rho \in \Sigma {(1)}$ we denote by $u_\rho \in N$ the minimal generator of $\rho \cap N$. Assuming that $D$ is ample, we consider the associated polytope $P=P_D\subset M_\mathbb {R}$:

(2.2)\begin{equation} P = \{m\in M_\mathbb{R}\ : \ \langle m,u_\rho\rangle \geq{-}a_\rho \text{ for all } \rho\in \Sigma(1)\} . \end{equation}

Note that if $D^{\prime }$ is equivariant and linearly equivalent to $D$ then $P_{D^{\prime }}$ is given by translation of $P_D$ in $M$ by some lattice element $m\in M$. In the same way, a lattice translation of the polytope $P_D$ corresponds to a different linearization of the action of $T$ on the line bundle $\mathcal {O}(D)$ (see § 3.1).

As $D$ is ample, and $P$ has normal fan equal to $\Sigma$, we have a correspondence between cones in $\Sigma$ and faces of $P$. For a face $Q$ in $P$, we denote by $\sigma _Q\in \Sigma$ (resp. by $O(Q)$) the associated cone (resp. the associated orbit). In particular, rays in $\Sigma$ corresponds to facets of $P$. For each $\rho \in \Sigma (1)$ the associated facet is

\[ F = P\cap \{m\in M_\mathbb{R}\ : \ \langle m,u_\rho\rangle ={-}a_\rho\}. \]

We will denote $u_\rho$ by $u_F$ and $a_\rho$ by $a_F$. We can also write

\[ D =\sum_{F \prec P} a_F D_F, \]

where the sum is over all facets of $P$ and $D_F:=D_{\rho _F}$. We will denote faces of $P$ of higher codimension by the letter $Q$ and vertices by the letter $v$. We use the relation $Q_1\preccurlyeq Q_2$ to signify that $Q_1$ and $Q_2$ are faces of $P$, possibly equal to $P$ itself, and $Q_1\subseteq Q_2$.

The correspondence between polarizations on $X$ and lattice polytopes with normal fan $\Sigma$ modulo lattice translations extends to real ample classes. We recall the definition of real ample divisors on a normal complex algebraic variety $X$ (see [Reference Lazarsfeld15]). Let $\mathrm {Div}(X)$ denote the group of integral Cartier divisors on $X$. The Néron-Severi group is given by $N^{1}(X) =\mathrm {Div}(X)/\sim _{num}$ and we set $N^{1}(X)_\mathbb {R}=N^{1}(X)\otimes _\mathbb {Z}\mathbb {R}$. Denote by $\mathrm {WDiv}(X)$ the set of Weil divisors on $X$ and $\mathrm {WDiv}_\mathbb {R}(X)$ the vector space of real Weil divisors.

We then have the following:

Proposition 2.2 Let $X$ be a projective toric variety given by a fan $\Sigma$. Then there is a bijective correspondence between real ample classes on $X$ and real polytopes of the form

\[ P=\lbrace m\in M_\mathbb{R}\ :\ \langle m, u_\rho \rangle \geq{-} a_\rho,\text{ for all } \rho\in \Sigma(1)\rbrace \]

for which the normal fan $\Sigma _P=\Sigma,$ modulo real translations within $M_\mathbb {R}$.

Proof. This statement between integral classes and lattice polytopes is standard (note that for projective toric varieties coming from fans, the real Picard group and the real Néron-Severi group coincide, see [Reference Cox, Little and Schenck2, proposition 6.3.15]). The rational case follows by clearing denominators and scaling polytopes. We now prove the real case. Set $\mathrm {Pic}(X)$ and $\mathrm {Cl}(X)$ the Picard and class groups of $X$. From the exact sequence (see [Reference Cox, Little and Schenck2, theorem 4.1.3]):

\[ 0\longrightarrow M\longrightarrow \bigoplus_{\rho\in\Sigma(1)}\mathbb{Z}\cdot D_\rho\stackrel{\pi}{\longrightarrow}\mathrm{Cl}(X)\longrightarrow 0 \]

we deduce the sequences of vector spaces, for $\mathbb {K}=\mathbb {Q}$ or $\mathbb {R}$:

\[ 0 \longrightarrow M_\mathbb{K}\longrightarrow W_\mathbb{K}\longrightarrow \mathrm{Pic}(X)_\mathbb{K}\longrightarrow 0 \]

where we set $W=\pi ^{-1}(\mathrm {Pic}(X))$, $W_\mathbb {K}=W\otimes _\mathbb {Z}\mathbb {K}$ and $\mathrm {Pic}(X)_\mathbb {K}=\mathrm {Pic}(X)\otimes _\mathbb {Z}\mathbb {K}$.

Let $\alpha \in \mathrm {Pic}(X)_\mathbb {R}= N^{1}(X)_\mathbb {R}$ be an ample class. We can represent $\alpha$ by

\[ D=\sum_{\rho\in\Sigma(1)} a_\rho D_\rho\in W_\mathbb{R}. \]

Define the set

\[ P_\alpha= \{m\in M_\mathbb{R}\ :\ \forall\rho\in\Sigma(1),\ \langle m,u_\rho\rangle\geq{-}a_\rho\}. \]

First observe that $P_\alpha$ is a polytope, rather than a polyhedron, since the fan $\Sigma$ is complete. By definition of ample real divisors, $D=\sum _{i=1}^{N}\alpha _iD_i$ for $D_i$ ample Cartier divisors and $\alpha _i$ positive real numbers. By the openness of the ampleness condition, we can take positive rational numbers $\beta _i\in \mathbb {Q}$ as close as we like to the $\alpha _i\in \mathbb {R}$ in such a way that

\[ D_\beta=\sum_{i=1}^{N} \beta_i D_i=\sum_{\rho\in\Sigma(1)} b_\rho D_\rho \]

is an ample $\mathbb {Q}$-divisor, and hence the polytope

\[ P_{\beta}=\{m\in M_\mathbb{R}\ :\forall\rho\in\Sigma(1),\ \langle m,u_\rho\rangle \geq{-}b_\rho\} \]

has normal cone $\Sigma _{P_\beta }=\Sigma$. Furthermore, decomposing the divisors $D_i$ in the basis $\lbrace D_\rho \, :\, \rho \in \Sigma (1) \rbrace$, we see that the values $(b_\rho )$ vary continuously with respect to $(\beta _i)$, so can be made sufficiently close to the $(a_\rho )$ to guarantee that $\Sigma _{P_\alpha }=\Sigma _{P_\beta }=\Sigma$. Note that as in the integral case, translations of $P_\alpha$ by elements of $M_\mathbb {R}$ correspond to different choices of representative of the class $\alpha$ in $W_\mathbb {R}$.

For the converse statement, we consider the polytope in $M_\mathbb {R}$

\[ P=\{m\in M_\mathbb{R}\,:\,\forall\rho\in\Sigma(1), \langle m,u_\rho\rangle \geq{-}a_\rho\} \]

for $a_\rho \in \mathbb {R}$, supposing that the normal fan of $P$ determined by the vectors $u_\rho$ is the fan of $X$. Then, the polytope $P$ determines the real Weil divisor $D_P=\sum _\rho a_\rho D_\rho \in \mathrm {WDiv}_\mathbb {R}(X)$. We show that $D_P$ lies in the space of real Cartier divisors, and is moreover ample. This is proven by induction on the number of $a_\rho$'s that are irrational. We list the rays in $\Sigma$ by $\rho _1,\,\ldots,\,\rho _d$ for $d=\#\Sigma (1)$. As noted above, the case where all $a_\rho$'s are rational is well-known. Suppose that, for fixed $k\geq 1$, $D=\sum _{i=1}^{d} a_{\rho _i}D_{\rho _i}$ defines a real ample class whenever its normal fan $\Sigma _P=\Sigma$ and $a_{\rho _i}\in \mathbb {Q}$ for all $i\geq k$. If $a_{\rho _i}\in \mathbb {Q}$ for $i\geq k+1$ then let $r_1,\,r_2$ be rational numbers for which $r_1< a_{\rho _k}< r_2$ sufficiently close to $a_{\rho _k}$ that for any $s\in [r_1,\,r_2]$ the polytope defined by the inequalities

\begin{align*} \langle m,u_{\rho_i}\rangle & \geq{-}a_{\rho_i},\text{ for } i\neq k,\\ \langle m,u_{\rho_k} \rangle & \geq{-}s \end{align*}

defines the same normal fan $\Sigma$. Then, for some $t\in [0,\,1]$, we have $a_{\rho _k}=tr_1 +(1-t)r_2$ and

\[ D=\sum_{i=1}^{d}a_{\rho_i}D_{\rho_i} = t\left( r_1 D_{\rho_k}+\sum_{i\neq k} a_{\rho_i}D_{\rho_i} \right) +(1-t)\left( r_2D_{\rho_k}+\sum_{i\neq k}a_{\rho_i}D_{\rho_i}\right). \]

By the induction hypothesis, each of the two real divisors on the right-hand side of the above equality is ample. By the convexity of the set of real ample classes, $D$ is ample.

2.2 Equivariant reflexive sheaves

We refer to [Reference Klyachko12, Reference Kool14, Reference Perling20] for this section. We consider a projective toric variety $X$ together with a polytope $P\subset M_\mathbb {R}$ associated to an ample class on $X$. We are interested in a combinatorial description of equivariant sheaves on $X$.

Recall that a reflexive sheaf on $X$ is a coherent sheaf $\mathcal {E}$ that is canonically isomorphic to its double dual $\mathcal {E}^{\vee \vee }$. Klyachko gave a description of equivariant reflexive sheaves in terms of combinatorial data:

To a family of filtrations $\mathbb {E}:=\lbrace (E^{F}(i))\subseteq E\,:\, F\prec P,\, i\in \mathbb {Z}\rbrace$ we can assign an equivariant reflexive sheaf $\mathcal {E}:=\mathfrak {K}(\mathbb {E})$ defined by

(2.3)\begin{equation} \Gamma(U_{\sigma_Q}, \mathcal{E}):=\bigoplus_{m\in M} \bigcap_{Q\preccurlyeq F} E^{F}(\langle m,u_F\rangle)\otimes \chi^{m} \end{equation}

for all proper faces $Q\prec P$, while $\Gamma (U_{\sigma _P},\,\mathcal {E})=E\otimes \mathbb {C}[M]$.

The morphisms of families of filtrations are defined by:

Such a morphism between families of filtrations induces an equivariant morphism between the associated reflexive sheaves.

Remark 2.9 In this paper, by a morphism between equivariant reflexive sheaves $\mathcal {E}_1=\mathfrak {K}(\mathbb {E}_1)$ and $\mathcal {E}_2=\mathfrak {K}(\mathbb {E}_2)$, we mean an equivariant morphism of degree zero: the space $\mathrm {Hom}(\mathcal {E}_1,\,\mathcal {E}_2)$ of all sheaves homomorphisms is naturally graded over $M$:

\[ \mathrm{Hom}(\mathcal{E}_1,\mathcal{E}_2)\subset \mathrm{Hom}(E_1,E_2)\otimes\mathbb{C}[M] \]

and the morphisms that we will consider are those lying in the $\mathrm {Hom}(E_1,\,E_2)\otimes \chi ^{0}$-component of the grading. In particular, this excludes injective maps like $\mathcal {O}_{\mathbb {P}^{1}}(-1)\to \mathcal {O}_{\mathbb {P}^{1}}$.

For a toric variety $Z$, and an ample polytope $P_Z$, we denote by:

1. i) $\mathfrak {Ref}^{T}(Z)$ the category of equivariant reflexive sheaves on $Z$,

2. ii) $\mathfrak {Filt}(P_Z)$ the category of families of filtrations associated to $P_Z$.

From Klyachko and Perling [Reference Klyachko12, Reference Perling20], we obtain the following:

As the category of filtrations on a given finite dimensional vector space is abelian, we have:

We will need the combinatorial characterizations of equivariant reflexive saturated subsheaves and of equivariant rank $1$ reflexive sheaves. Let $\mathcal {E}=\mathfrak {K}(\mathbb {E})$ be an equivariant reflexive sheaf on $X$, given by a family of filtrations $\mathbb {E}=\lbrace (E^{F}(i))\subset E: F\prec P,\, i\in \mathbb {Z}\rbrace$. For any vector subspace $W\subset E$, define a family of filtrations $\mathbb {E}\cap W$ by

\[ \mathbb{E}\cap W = \lbrace (W\cap E^{F}(i))\subset W\cap E: F\prec P, i\in\mathbb{Z}\rbrace. \]

Then, the sheaf $\mathcal {E}_W:=\mathfrak {K}(\mathbb {E}\cap W)$ is an equivariant saturated reflexive subsheaf of $\mathcal {E}$. Any equivariant reflexive saturated subsheaf of $\mathcal {E}$ arises that way:

As for rank $1$ reflexive sheaves, from the definition we obtain:

Proposition 2.13 Let $\mathcal {O}(-D)$ be the rank $1$ reflexive sheaf associated to the invariant Weil divisor $D=\sum _{F\prec P} a_F D_F$. Then, $\mathcal {O}(-D)=\mathfrak {K}(\mathbb {E}_D),$ where the family of filtrations $\mathbb {E}_D=\lbrace (E^{F}(i))\subset \mathbb {C},\, F\prec P,\, i\in \mathbb {Z}\rbrace$ satisfies

\[ E^{F}(i)= \left\{ \begin{array}{@{}ccc@{}} 0 & \mathrm{ if } & i < a_F\\ \mathbb{C} & \mathrm{ if } & i \geq a_F. \end{array} \right. \]

We will also need the determinant and first Chern class of reflexive sheaves.

Remark 2.14 Let $A_k(X)$ be the $k$-th Chow group of $X$. This is the quotient of the free abelian group on $k$-dimensional subvarieties by rational equivalence. The first Chern class is the map $c_1:\mathrm {Pic}(X)\to A_{n-1}(X)$ induced by the inclusion of Cartier divisors in Weil divisors. This defines a morphism $A_k(X)\to A_{k-1}(X)$ for each $k$ as follows. For $\mathcal {L}$ a line bundle and $V$ a $k$-dimensional subvariety on $X$, $\mathcal {L}|_V$ defines a Cartier divisor on $V$, hence a $(k-1)$-cycle on $X$.

As $X$ is normal, this definition extends to rank one reflexive sheaves since every rank-one reflexive sheaf is of the form $\mathcal {E}=\mathcal {O}_X(D_\mathcal {E})$ for some Weil divisor $D_\mathcal {E}$, and hence $c_1(\mathcal {E}):=[D_\mathcal {E}]\in A_{n-1}(X)$. If $H$ is an ample line bundle on $X$, the degree of $\mathcal {E}$ is then given by

\[ \mathrm{deg}_H(\mathcal{E})= c_1(\mathcal{E})\cdot H^{n-1}\in A_0(X)\cong \mathbb{Z}. \]

Recall the following:

We prove now that the previously known formula for the first Chern class for equivariant reflexive sheaves holds in our case, as a linear combination of Weil divisors. This extends the previous case on smooth varieties, given by Kool, to normal varieties. To produce a necessary combinatorial formula, we first need to introduce some notation:

Definition 2.16 Let $\mathcal {E}=\mathfrak {K}(\mathbb {E})$, with $\mathbb {E}=\lbrace (E^{F}(i))\subset E :F\prec P,\, i\in \mathbb {Z} \rbrace$. We set, for all $F\prec P$ and all $i\in \mathbb {Z}$:

(2.4)\begin{equation} e^{F}(i)=\dim(E^{F}(i))-\dim (E^{F}(i-1)). \end{equation}

We will refer to the integers $(e^{F}(i))_{F\prec P, i\in \mathbb {Z}}$ as the dimension jumps of $\mathbb {E}$ or $\mathcal {E}$.

Then we have:

Proposition 2.17 Let $\mathcal {E}=\mathfrak {K}(\mathbb {E})$ be a rank $r$ equivariant reflexive sheaf on $X,$ given by a family of filtrations $\mathbb {E}=\lbrace (E^{F}(i))\subset E: F\prec P,\, i\in \mathbb {Z}\rbrace$. We define a family of filtrations $\det (\mathbb {E})=\lbrace (E_{\det }^{F}(i))\subset \Lambda ^{r} E: F\prec P,\, i\in \mathbb {Z}\rbrace$ by setting, for all $F\prec P,$

\[ E_{\det}^{F}(i)= \left\{ \begin{array}{@{}ccc@{}} 0 & \mathrm{ if } & i < i_F(\det\mathcal{E})\\ \Lambda^{r} E & \mathrm{ if } & i \geq i_F(\det\mathcal{E}) \end{array} \right. \]

where for all $F\prec P,$

\[ i_F(\det\mathcal{E})=\sum_{i\in\mathbb{Z}} i e^{F}(i). \]

Then $\det (\mathcal {E})=\mathfrak {K}(\det (\mathbb {E}))$.

Proof. Note first that because $\mathfrak {Ref}^{T}(X)$ is abelian, $\Lambda ^{r}\mathcal {E}$ is reflexive and $\det (\mathcal {E})=\Lambda ^{r} \mathcal {E}$. Then, the family of filtrations $\lbrace (\Lambda ^{r} E^{F}(i)): F\prec P,\, i\in \mathbb {Z} \rbrace$ for $\Lambda ^{r} \mathcal {E}$ satisfies:

\[ \Lambda^{r} E^{F}(i) = \sum_{i_1,\ldots, i_r: \sum i_j = i} E^{F}(i_1) \wedge \ldots \wedge E^{F}(i_r). \]

Now, $\Lambda ^{r} E$ is one dimensional, so

\[ \Lambda^{r} E^{F}(i)= \left\{ \begin{array}{@{}ccc@{}} 0 & \mathrm{ if } & i < j_F\\ \Lambda^{r} E & \mathrm{ if } & i \geq j_F \end{array} \right. \]

where $j_F$ is the smallest integer $l\in \mathbb {Z}$ such that there is a partition $i_1,\,\ldots,\,i_r$ of $l$ with $E^{F}(i_1) \wedge \ldots \wedge E^{F}(i_r)\neq \lbrace 0 \rbrace$. From the fact that $(E^{F}(i))$ forms a filtration of vector spaces, we deduce that $j_F$ must be the sum of the integers $i$ such that the dimension of $E^{F}(i)$ changes, counted with multiplicity. Hence $j_F = \sum _{i\in \mathbb {Z}} i e^{F}(i)$, which ends the proof.

Corollary 2.18 Let $\mathcal {E}=\mathfrak {K}(\mathbb {E})$ be an equivariant reflexive sheaf on $X,$ given by a family of filtrations $\mathbb {E}=\lbrace (E^{F}(i))\subset E:F\prec P,\, i\in \mathbb {Z}\rbrace$. The first Chern class of $\mathcal {E}$ is the class of the Weil divisor:

(2.5)\begin{equation} c_1(\mathcal{E})={-}\sum_{F\prec P} i_F(\det\mathcal{E}) \: D_F. \end{equation}

where for all $F\prec P$,

\[ i_F(\det\mathcal{E})=\sum_{i\in\mathbb{Z}} i e^{F}(i). \]

3.1 Toric GIT

We refer to [Reference Mumford, Fogarty and Kirwan17, Reference Thaddeus22] for this section. We are interested in GIT quotients of $(X,\,L)$ by subtorus actions. Let $N_0$ be a sublattice of $N$ of rank $g$. We will assume that $N_0$ is saturated, that is $N_0=(N_0\otimes _\mathbb {Z} \mathbb {R})\cap N$. The sublattice $N_0$ spans a $g$-dimensional subtorus $G=N_0\otimes _\mathbb {Z} \mathbb {C}^{*}$ of $T$. We fix a linearization $\gamma$ of $T$ on $L$ and we will consider the GIT quotient with respect to the induced linearization of $G$. From § 2.1, $(X,\,L)$ defines a family of lattice polytopes $\lbrace P_D:\mathcal {O}(D)\sim L\rbrace$, all equal up to translations by lattice elements. Then, the linearization $\gamma$ determines a unique polytope $P$ in this family such that

(3.1)\begin{equation} H^{0}(X,L)=\bigoplus_{m\in P\cap M} \mathbb{C}\cdot \chi^{m} \end{equation}

is the weight decomposition of the $T$-action on the space of global sections induced by $\gamma$ (see [Reference Cox, Little and Schenck2, proposition 4.3.3]). We have the following:

\[ 0 \to N_0 \to N \to N/N_0 \to 0, \]

and the associated dual sequence:

\[ 0 \to N_0^{{\perp}} \to M \to M/N_0^{{\perp}} \to 0. \]

Set $U=N_0^{\perp }\otimes _\mathbb {Z} \mathbb {R}\subset M_\mathbb {R}$. Then from [Reference Thaddeus22, proposition 3.2]:

We can also describe the stable and semistable points :

From this proposition, the set of faces of the polytope of $X^{ss}/\!\!/G$ is

\[ \lbrace U\cap Q: Q\preccurlyeq P\rbrace. \]

We will denote by $P^{s}$ (resp. $P^{ss}$ and $P^{us}$) the set of faces corresponding to stable orbits (resp. semistable orbits and unstable orbits) of $X$. For technical reasons, we will need the following assumption on the action:

The nice fact about generic actions is the following:

From now on, unless explicitly stated, we will assume that $(G,\,\gamma )$ is generic. We end this section with some definitions and lemmas relating the primitive normals to the facets of $P$ to those of $P_Y$. We consider the lattice projection, $\pi :N\to N_Y=N/N_0$.

Lemma 3.8 There exists a unique positive integer $b_F$ such that

\[ \check{u}_F=\frac{1}{b_F} \pi(u_F). \]

Let $F$ be facet in $P^{s}$. As $u_F$ is primitive, we can complete it into a basis $\mathcal {B}_F:=\lbrace u_F,\,u_i : \: i= 2\ldots n \rbrace$ of $N$. We denote by $\mathcal {B}_F^{*}=\lbrace m_F,\, m_i : \: i=2\ldots n \rbrace$ the dual basis of $M$. In particular, note that $\lbrace m_i:\: i=2\ldots n \rbrace$ is a basis for $u_F^{\perp }\cap M$. Similarly, we set $\check{\mathcal {B}}_F:= \lbrace \check{u}_F,\, \check{u}_i\,: \: i= 2\ldots n-g \rbrace$ a basis for $N_Y$ with dual basis $\check{\mathcal {B}}_F^{*}:= \lbrace \check{m}_F,\, \check{m}_i: \: i= 2\ldots n-g \rbrace$ and again $\lbrace \check{m}_i:\: i=2\ldots n-g \rbrace$ is a basis for $\check{u}_F^{\perp }\cap N_Y$. From lemma 3.8, we deduce:

Lemma 3.9 The element $\check{m}_F\in N_0^{\perp }\subset M$ can be uniquely written as

\[ \check{m}_F= b_F m_F + m_F^{{\perp}} \]

with $m_F^{\perp }\in u_F^{\perp }$.

We conclude this section with the following observation:

3.2 Descent criteria for equivariant reflexive sheaves

We retain the notation from the previous section. We assume as before that $(X,L)$ is a polarized toric variety of dimension $n$ and that $(G,\,\gamma )$ is generic. We consider $(Y,\,\check{L})$ the GIT quotient associated to the $G$ action on $(X,\,L)$. Following [22] and the discussion in the previous section, we recall that $Y$ is a simplicial toric variety, with torus given by $T_Y=T_X/G$, and $\check{L}$ is the invariant ample line bundle on $Y$ corresponding to the polytope $P_Y=P\cap U$. We wish to compare equivariant reflexive sheaves on $X$ and equivariant reflexive sheaves on $Y$. We denote by $\iota : X^{s} \to X$ the inclusion and by $\pi : X^{s} \to Y$ the projection. We then introduce:

Let $\mathcal {E}$ be a $G$-equivariant coherent sheaf on $X$. For simplicity we will denote by $\mathcal {E}^{s}$ the restriction $\iota ^{*}\mathcal {E}$. As explained for example in [Reference Nevins19], $\mathcal {E}$ descends to $Y$ if and only if

\[ \mathcal{E}^{s}\simeq \pi^{*} \pi^{G}_* \mathcal{E}^{s} \]

where $\pi ^{G}_*$ is the invariant pushforward.

We will give a description of the functors $\iota ^{*}$, $\pi ^{*}$ and $\pi ^{G}_*$ for torus-equivariant reflexive sheaves in terms of families of filtrations. Making use of the equivalence of categories $\mathfrak {K}$, by abuse of notations, we will use the same symbols to design the associated functors on families of filtrations.

Proof. Let $F\prec P^{s}$. Then we have the GIT quotient projection

\[ \pi : U_{\sigma_F}=\mathrm{Spec}(\mathbb{C}[\sigma_F^{{\vee}}\cap M]) \to U_{\sigma_{F \cap U}}=\mathrm{Spec}(\mathbb{C}[\sigma_{U\cap F}^{{\vee}}\cap N_0^{{\perp}}]) \]

and by definition,

\[ \Gamma(U_{\sigma_{F \cap U}},\pi_*^{G}\mathcal{E}^{s})=\bigoplus_{m\in N_0^{{\perp}}} \Gamma(U_{\sigma_F},\mathcal{E}^{s})_m. \]

Thus for all $m\in N_0^{\perp }$:

\[ \Gamma(U_{\sigma_{F \cap U}},\pi_*^{G}\mathcal{E}^{s})_m= \Gamma(U_{\sigma_F},\mathcal{E}^{s})_m \]

that is

\[ \check E^{U\cap F}(\langle m,\check{u}_F\rangle)\otimes \chi^{m} = E^{F}(\langle m, u_F\rangle)\otimes \chi^{m}. \]

Using lemma 3.9 and the basis $\check{ \mathcal {B}}_F^{*}$ and $\mathcal {B}_F^{*}$ to decompose $m$ we obtain the result.

Similarly, we have

In particular, the dimension jump values $e^{F}(i)$ can only be non-zero for values of $i$ that are multiples of $b_F$.

Proof. By lemma 3.14, we know that $E^{F}(b_Fi)=\check E^{U\cap F}(i)$ for $i\in \mathbb {Z}$, and hence we also have $E=\check E$. In general, we have for $j,\,i\in \mathbb {Z}$:

\[ \check E^{U\cap F}(j)\otimes \chi^{j \check m_F}=\Gamma(U_{\sigma_{F\cap U}},\quad \check{ \mathcal{E}})_{j\check m_F} \]

and

\[ E^{U\cap F}(i)\otimes \chi^{i m_F}=\Gamma(U_{\sigma_F}, \pi^{*}\check{\mathcal{E}})_{i m_F}. \]

Moreover, we have by definition

\[ \Gamma(U_{\sigma_F},\pi^{*}\check{ \mathcal{E}})=\Gamma(U_{\sigma_{F\cap U}}, \check{\mathcal{E}})\otimes_{\mathbb{C}[\sigma_{U\cap F}^{{\vee}}\cap N_0^{{\perp}}]} \mathbb{C}[ \sigma_F^{{\vee}}\cap M]. \]

Note that $m\in M\cap \sigma _F^{\vee }$ if and only if $\langle m,\,u_F \rangle \geq 0$ and $m\in N_0^{\perp }\cap \sigma _{U\cap F}^{\vee }$ if and only if $\langle m,\,\check{u}_F\rangle \geq 0$. Thus, to prove the lemma, it is enough to show that if $im_F=m'+(i m_F - m')$ with $m'\in N_0^{\perp }\cap \sigma _{U\cap F}^{\vee }$, and $(im_F -m')\in M\cap \sigma _F^{\vee }$ then $\langle m',\,\check{u}_F\rangle \leq \lfloor \frac {i}{b_F}\rfloor$ (note that we use the fact that the spaces $(E^{F}(i))$ form a filtration). Suppose then that we have such a decomposition $im_F=m'+(i m_F - m')$ for $i m_F$. Without loss of generality, we can assume that $m'=a \check m_F$ with $a=\langle m',\,\check{u}_F\rangle \in \mathbb {N}$. Then we have $\langle i m_F - a \check m_F,\,u_F\rangle \geq 0$ and thus by lemma 3.9 we obtain $i- a b_F \geq 0$. The result follows.

From these lemmas we deduce a version of Nevins’ criterion for descent of equivariant reflexive sheaves on toric varieties [Reference Nevins19]:

Proof. The proof follows from the fact that $\mathcal {E}$ descends if and only if

\[ \iota^{*}\mathbb{E}\simeq \pi^{*} \pi^{G}_* \iota^{*}\mathbb{E}. \]

Using lemmas 3.14 and 3.15, this is equivalent that for all $F\prec P^{s}$, for all $i\in \mathbb {Z}$,

\[ E^{F}(i)= E^{F}\left(b_F \left\lfloor\frac{i}{b_F}\right\rfloor\right). \]

The result follows.

For later use, we emphasize the following corollary:

3.3 Pullback functors

We preserve notation of the previous section, still assuming the pair $(G,\,\gamma )$ to be generic. In this section, we introduce pull-back functors from $\mathfrak {Ref}^{T}(Y)$ to $\mathfrak {Ref}^{T}(X)$. Note that the superscript $T$ stands for equivariant, and means $T_X$-equivariant on $X$, and $T_Y$-equivariant on $Y$. The images of these functors will contain precisely the equivariant reflexive sheaves on $X$ that descend to $Y$ and that are suitable to compare slope stability notions on $X$ and $Y$.

Let $\mathbf {D}^{u}:=\lbrace D_F:\: F \prec P^{us} \rbrace$ be the set of unstable irreducible $T$-equivariant divisors on $X$. For ${\bf i}:=(i_D)_{D\in \mathbf {D}^{u}}\in \mathbb {Z}^{u}$, we define a functor $\mathfrak {P}_{\bf {i}}$ from $\mathfrak {Ref}^{T}(Y)$ to $\mathfrak {Ref}^{T}(X)$. Using the functor $\mathfrak {K}$, it is sufficient to define the corresponding functor from $\mathfrak {Filt}(P_Y)$ to $\mathfrak {Filt}(P)$. Let $\check{ \mathbb {E}}\in \mathfrak {Filt}(P_Y)$, with $\check{ \mathbb {E}}=\lbrace (\check E^{\check F}(i))\subset \check E: \check F\prec P_Y,\, i\in \mathbb {Z} \rbrace$. Then we define $\mathfrak {P}_\mathbf {i}(\check{\mathbb {E}})=\lbrace (E^{F}(i))\subset E:F\prec P,\, i\in \mathbb {Z} \rbrace$ by setting $E =\check E$ and for $F\prec P$ and $i\in \mathbb {Z}$:

• • If $F\prec P^{s}$, then $E^{F}(i)=\check E^{U\cap F}(\lfloor \frac {i}{b_F}\rfloor )$.

• • If $F\prec P^{us}$, then

  \[ E^{F}(i)= \left\{ \begin{array}{@{}ccc@{}} 0 & \mathrm{ if } & i < i_{D_F}\\ E & \mathrm{ if } & i \geq i_{D_F} \end{array} \right. . \]

We now define the functors $\mathfrak {P}_\mathbf {i}$ on morphisms of families of filtrations. Let $\check{\mathbb {E}}_1,\, \check{\mathbb {E}}_2 \in \mathfrak {Filt}(P_Y)$, with $\check{\mathbb {E}}_j=\lbrace (\check E_j^{\check F}(i))\subset \check E_j: \check F\prec P_Y,\, i\in \mathbb {Z} \rbrace$, for $j\in \lbrace 12 \rbrace$. Let $\check f\in \mathrm {Hom}(\check{\mathbb {E}}_1,\,\check{\mathbb {E}}_2)$, and recall that $\check f$ corresponds to a linear map $\check f: \check E_1 \to \check E_2$ such that for all $\check F\prec P_Y$ and $i\in \mathbb {Z}$, $\check f(\check E_1^{\check F}(i))\subset \check E_2^{\check F}(i)$. We then simply define $\mathfrak {P}_\mathbf {i}(\check f):=f$ to be the element of $\mathrm {Hom}(\mathfrak {P}_\mathbf {i}(\check{\mathbb {E}}_1),\,\mathfrak {P}_\mathbf {i}(\check{\mathbb {E}}_2))$ given by the linear map

\[ f=\check f : E_1=\check E_1 \to E_2=\check E_2. \]

We then check that $\mathfrak {P}_\mathbf {i}(\check f)$ is well defined, that is for all $F\prec P$ and $i\in \mathbb {Z}$,

\[ f(E_1^{F}(i))\subset E_2^{F}(i). \]

If $F\prec P^{us}$, this is clear as for $i < i_{D_F}$, $E_1^{F}(i)=E_2^{F}(i)=0$, while for $i\geq i_{D_F}$, $E_1^{F}=E_1$ and $E_2^{F}=E_2$. If $F\prec P^{s}$, then $E_j^{F}(i)=\check E_j^{U\cap F}(\lfloor \frac {i}{b_F}\rfloor )$, $j\in \lbrace 12 \rbrace$. However, by the compatibility of $\check f=f$ with the filtrations, for any $i'\in \mathbb {Z}$, we have

\[ \check f(\check E_1^{U\cap F}(i'))\subset \check E_2^{U\cap F}(i'), \]

which implies $f(E_1^{F}(i))\subset E_2^{F}(i)$. Thus $f$ is compatible with the families of filtrations and $\mathfrak {P}_\mathbf {i}(\check f)$ is well defined.

By abuse of notation, we denote by $\mathfrak {P}_\mathbf {i}$ the associated functors on $\mathfrak {Ref}^{T}(Y)$. We then have:

Proof. We keep the notations used to define $\mathfrak {P}_\mathbf {i}$. We need to show that the map

\[ \mathfrak{P}_\mathbf{i} : \mathrm{Hom}(\check{\mathbb{E}}_1,\check{\mathbb{E}}_2) \to \mathrm{Hom}(\mathfrak{P}_\mathbf{i}(\check{ \mathbb{E}}_1),\mathfrak{P}_\mathbf{i}(\check{ \mathbb{E}}_2)) \]

is a bijection. Injectivity is clear by definition of $\mathfrak {P}_\mathbf {i}(\check f)$. For surjectivity, let $f\in \mathrm {Hom}(\mathfrak {P}_\mathbf {i}(\check{\mathbb {E}}_1),\,\mathfrak {P}_\mathbf {i}(\check{ \mathbb {E}}_2))$. Then $f$ is described by a linear map from $E_1$ to $E_2$ that is compatible with the families of filtrations $\mathfrak {P}_\mathbf {i}(\check{ \mathbb {E}}_1)$ and $\mathfrak {P}_\mathbf {i}(\check{ \mathbb {E}}_2)$. By definition of $\mathfrak {P}_\mathbf {i}$, for $F\prec P^{us}$, the condition $f(E_1^{F}(i))\subset E_2^{F}(i)$ is satisfied for any linear map from $E_1$ to $E_2$. However, for $F\prec P^{s}$, the compatibility gives for all $i\in \mathbb {Z}$:

\[ \displaystyle f\left(\check E_1^{U\cap F}\left(\left\lfloor\frac{i}{b_F}\right\rfloor\right)\right)\subset \check E_2^{U\cap F}\left(\left\lfloor\frac{i}{b_F}\right\rfloor\right). \]

Taking in particular $i\in b_F\mathbb {Z}$, and using the fact that any facet of $P_Y$ is of the form $F\cap U$ for some facet $F\prec P^{s}$, we deduce that $f$ is a linear map between $\check E_1$ and $\check E_2$ that is compatible with the families of filtrations $\check{ \mathbb {E}}_1$ and $\check{ \mathbb {E}}_2$. Thus $f=\mathfrak {P}_\mathbf {i}(\check f)$ for $\check f = f$, seen as a linear map from $\check E_1=E_1$ to $\check E_2=E_2$.

Remark 3.19 There is a simple relation between the different functors $\mathfrak {P}_\mathbf {i}$. Given an equivariant reflexive sheaf $\mathcal {E}$ on $Y$, for any ${\bf i}=(i_D)_{D\in \mathbf {D}^{u}}\in \mathbb {Z}^{u}$, one has

\[ \mathfrak{P}_\mathbf{i}(\mathcal{E})=\mathfrak{P}_{\bf 0}(\mathcal{E}) \otimes \mathcal{O}_X\left(-\sum_{D\in \mathbf{D}^{u}} i_D D\right). \]

As will appear clearly in the next section, the difference of slopes between two reflexive sheaves is not affected by tensoring both sheaves by $\mathcal {O}_X(-\sum _{D\in \mathbf {D}^{u}} i_D D)$. Thus, from the point of view of moduli spaces of slope stable sheaves, one is reduced to the study of the functor $\mathfrak {P}_{\bf 0}$. We thank the referee for this interesting remark.

We give a geometric interpretation of the images of the pull-back functors. First, we interpret the condition on unstable divisors with the following lemma, whose proof is straightforward from the definitions.

We deduce:

It will become clear in the next section that the extension condition (3.2) for unstable divisors is precisely the condition on $\mathfrak {P}_\mathbf {i}$ that enables us to compare the slopes of $\check{\mathcal {E}}\in \mathfrak {Ref}^{T}(Y)$ and of $\mathfrak {P}_\mathbf {i}(\check{\mathcal {E}})$ for all $\check{\mathcal {E}}\in \mathfrak {Ref}^{T}(Y)$. We finish this section with a more effective lemma that will be used in these comparisons:

4.1 Some stability notions

We recall now the notions of stabilities that we will consider, and state some propositions specific to the toric setting. We refer to [Reference Guenancia8, Reference Huybrechts and Lehn10] for the stability notions (see in particular [Reference Guenancia8, proposition 4.3] for the definition of slope on normal varieties) and to [Reference Lazarsfeld15] for the definition of the intersection of a Weil divisor with real Cartier divisors.

Definition 4.1 Let $\mathcal {E}$ be a torsion-free coherent sheaf on $X$. The degree of $\mathcal {E}$ with respect to an ample class $\alpha \in N^{1}(X)_\mathbb {R}$ is the real number obtained by intersection:

\[ \mathrm{deg}_\alpha(\mathcal{E})=c_1(\mathcal{E})\cdot \alpha^{n-1} \]

and its slope with respect to $\alpha$ is given by

\[ \mu_\alpha(\mathcal{E}) = \frac{\mathrm{deg}_\alpha(\mathcal{E})}{\mathrm{rk}(\mathcal{E})}. \]

A torsion-free coherent sheaf $\mathcal {E}$ is said to be $\mu$-semi-stable or slope semi-stable with respect to $\alpha$ if for any coherent subsheaf $0\neq \mathcal {F}\subsetneq \mathcal {E}$ with $0<\mathrm {rk}\, \mathcal {F}<\mathrm {rk}\,\mathcal {E}$, one has

(4.1)\begin{equation} \mu_\alpha(\mathcal{F})\leq \mu_\alpha(\mathcal{E}). \end{equation}

When strict inequality always holds, we say that $\mathcal {E}$ is $\mu$-stable. Finally, $\mathcal {E}$ is said to be $\mu$-polystable if it is the direct sum of $\mu$-stable subsheaves of the same slope.

It is a standard fact that to test stability of a torsion-free coherent sheaf $\mathcal {E}$, it is enough to check (4.1) for proper saturated coherent subsheaves $\mathcal {F}$ (see [Reference Huybrechts and Lehn10, proposition 1.2.6]). A less standard fact that follows from [Reference Kool14, proposition 4.13] is that in the toric setting, to check $\mu$-stability of an equivariant reflexive sheaf, it is enough to compare slopes for saturated equivariant reflexive subsheaves. More precisely, using the description of saturated equivariant reflexive subsheaves from proposition 2.12, we have:

Note that the proof of the above proposition is valid on normal toric varieties. Following [Reference Hering, Nill and Suess9, Reference Kool14], we will use the following combinatorial formula for the slopes of equivariant reflexive sheaves (see also [Reference Hering, Nill and Suess9, lemma 2.2]):

Lemma 4.3 Let $\mathcal {E}$ be a $T$-equivariant reflexive sheaf with dimension jumps $(e^{F}(i))$. Then

\[ \mu_L(\mathcal{E})={-}\frac{1}{\mathrm{rk}(\mathcal{E})}\sum_{F\prec P} i_F(\det\mathcal{E}) \mathrm{deg}_L(D_F), \]

where for all $F\prec P,$

\[ i_F(\det\mathcal{E})=\sum_{i\in\mathbb{Z}} i e^{F}(i). \]

We will need to compare the data defining the slopes after descent:

We will also need the degree for the pullbacks of the irreducible $T$-invariant divisors.

Lemma 4.5 Let $F_1\prec P^{s}$. Then for any $\mathbf {i}\in \mathbb {Z}^{u},$

\[ \mathfrak{P}_\mathbf{i}(\mathcal{O}_Y(D_{F_1\cap U}))=\mathcal{O}_X\left(b_{F_1}D_{F_1}-\sum_{F\prec P^{us}}i_FD_F\right). \]

Proof. The sheaf $\mathcal {O}_Y(D_{F_1\cap U})$ is equivariant and reflexive. Combining proposition 2.13 and the definition of the functor $\mathfrak {P}_\mathbf {i}$, $\mathfrak {P}_\mathbf {i}(\mathcal {O}_Y(D_{F_1\cap U}))$ is determined by the family of filtrations

\begin{align*} L^{F_1}(i)& = \begin{cases} 0 & i<{-}b_{F_1},\\ \mathbb{C} & i\geq{-}b_{F_1}, \end{cases} \\ L^{F}(i)& = \begin{cases} \begin{array}{@{}cc@{}} 0 & i<0,\\ \mathbb{C} & i\geq 0, \end{array} \text{ if } F\neq F_1 \text{ is stable},\\ \begin{array}{@{}cc@{}} 0 & i< i_F,\\ \mathbb{C} & i\geq i_F, \end{array} \text{ if } F \text{ is unstable}. \end{cases} \end{align*}

That is to say, $\mathfrak {P}_\mathbf {i}(\mathcal {O}_Y(D_{F_1\cap U}))=\mathcal {O}_X(b_{F_1}D_{F_1}-\sum _{F\prec P^{us}}i_FD_F)$.

Corollary 4.6 Let $F_1\prec P^{s}$. Then for any $\mathbf {i}\in \mathbb {Z}^{u},$

\[ \mathrm{deg}_L( \mathfrak{P}_\mathbf{i}(\mathcal{O}_Y(D_{F_1\cap U})))=b_{F_1} \mathrm{deg}_L(D_{F_1})-\sum_{F\prec P^{us}}i_F \mathrm{deg}_L(D_F). \]

4.2 Comparison of slope stability via the Minkowski condition

In this section, we prove theorem 4.7 and proposition 4.8, from which theorem 1.1 follows. Recall from § 2.1 and 3.1 that the data of $(X,\,L)$ together with the linearization $\gamma : T \to \mathrm {Aut}(L)$ determines a polytope $P \subset M_\mathbb {R}$, and that facets $F$ in the stable locus $P^{s}$ of $P$ correspond to stable irreducible $T$-invariant divisors $D_F$.

Theorem 4.7 Let $(X,\,L)$ be a polarized toric variety with torus $T=N\otimes _\mathbb {Z}\mathbb {C}^{*}$. Let $G=N_0\otimes _\mathbb {Z}\mathbb {C}^{*}$ be a subtorus for a saturated sublattice $N_0\subset N$. Let $\gamma : T\to \mathrm {Aut}(L)$ be a generic linearization of $G,$ and denote by $Y$ the associated GIT quotient $X/\!\!/G$. Assume that Minkowski condition holds:

(4.2)\begin{equation} \sum_{F\prec P^{s}} \mathrm{deg}_L(D_F)\: u_F=0 \mod N_0\otimes_\mathbb{Z}\mathbb{R}. \end{equation}

Then there exists a unique ample class $\alpha \in N^{1}(Y)_\mathbb {R}$ such that for every equivariant reflexive sheaf $\check {\mathcal {E}}$ on $Y$, for any $\mathbf {i}=(i_F)_{F\prec P^{us}}\in \mathbb {Z}^{u}$, setting $\mathcal {E}=\mathfrak {P}_\mathbf {i}(\check {\mathcal {E}})$, we have

(4.3)\begin{equation} \mu_L(\mathcal{E})=\mu_{\alpha}(\check{\mathcal{E}})-\sum_{F\prec P^{us}}i_F\mathrm{deg}_L(D_F). \end{equation}

In this case, $\mathcal {E}$ is stable on $(X,\,L)$ if and only if $\check {\mathcal {E}}$ is stable on $(Y,\,\alpha )$.

A converse of this statement is given in the following.

Proposition 4.8 There exists an ample class $\alpha \in N^{1}(Y)_\mathbb {R}$ with respect to which the functors $\mathfrak {P}_\mathbf {i}$ from equivariant reflexive sheaves on $Y$ to equivariant reflexive sheaves on $X$ preserve each of the conditions of $\mu$-stability, $\mu$-semi-stability and $\mu$-polystability if and only if

\[ \sum_{F \prec P^{s}}\mathrm{deg}_P(D_F)u_F=0 \mod\ N_0\otimes_\mathbb{Z}\mathbb{R}. \]

We delay the proofs slightly in order to first present the classical Minkowski theorem. We recall that a lattice $M$ defines a measure $\nu$ on $M_\mathbb {R}=M\otimes _\mathbb {Z}\mathbb {R}$ as the pull-back of Haar measure on $M_\mathbb {R}/M$. It is determined by the properties

1. (a) $\nu$ is translation invariant,

2. (b) $\nu (M_\mathbb {R}/M)=1$.

Let $\alpha \in N^{1}(X)_\mathbb {R}$ be an ample class, and denote by $P_\alpha$ the associated polytope. For any facet $F$ of $P_\alpha$, there is a unique $(n-1)$-form $\nu _F$, independent of $\alpha$, such that $\nu _F\wedge u_F = \nu$. If $P_\alpha$ is a lattice polytope, with

\[ P_\alpha=\lbrace m\in M_\mathbb{R} : \langle m , u_F\rangle\geq{-} a_F \textrm{ for all } F\prec P \rbrace, \]

the form $\nu _F$ corresponds to the measure defined by the lattice

\[ M\cap \lbrace m\in M_\mathbb{R} : \langle m , u_F\rangle={-} a_F \rbrace \]

in the affine span of $F$. We will denote the volume of the facet $F$ with respect to $\nu _F$ by ${\rm latvol}(F)$.

Proposition 4.9 [Reference Danilov3]

For any facet $F$ of $P_\alpha,$

\[ \mathrm{deg}_\alpha(D_F)=\mathrm{latvol}(F). \]

We have the elementary lemma:

Lemma 4.10 Let $u_F\in N$ be the primitive element associated to a facet $F\prec P$. Let $g=\langle \cdot,\,\cdot \rangle$ be a positive definite inner product on $M_\mathbb {R}$ whose induced volume form coincides with $\nu$ and let $\tilde {u}_F= u_F/\|u_F\|$ be the normalized vector in $N_\mathbb {R}$. Then,

\[ {\rm latvol}(F)\,u_F={\rm vol}_g(F)\,\tilde{u}_F. \]

Here ${\rm vol}_g$ denotes the $(n-1)$-dimensional Euclidean volume of the facet $F$, calculated with respect to the inner product $g$. From this observation we obtain the counterpart of a classical result of Minkowski, adapted to the case of lattice polytopes.

Proposition 4.11 Let $u_1,\,\ldots,\,u_r\in N$ be distinct primitive lattice elements that span the real vector space $N_\mathbb {R}$ and let $f_1,\,\ldots,\, f_r>0$ be positive real numbers. Then there exists a compact and convex polytope in $M_\mathbb {R}$ whose facets have inward normals the elements $(u_i)$ and lattice volumes the $(f_i)$ if and only if

\[ \sum_{i=1}^{r}f_iu_i=0. \]

Moreover, such a polytope is unique up to translation.

We deduce the following interesting corollary on intersection theory of toric varieties:

We can now give the proofs of theorem 4.7 and proposition 4.8.

Proof of theorem 4.7. Let $\mathbf {i}\in \mathbb {Z}^{u}$. Let $\check {\mathcal {E}}$ be an equivariant reflexive sheaf on $Y$ and let $\mathcal {E}=\mathfrak {P}_\mathbf {i}(\check {\mathcal {E}})$ on $X$, both of rank $r$. Let $\alpha$ be an ample $\mathbb {R}$-divisor on $Y$ that determines a real polytope $\check P\subseteq \check {M}_\mathbb {R}= N_0^{\perp }\otimes _\mathbb {Z}\mathbb {R}$ whose facets $\check {F}$ have primitive normals $\check{u}_F=u_{F\cap U}\in N/N_0$ equal to those of $P_Y$. Then, from lemma 4.3,

\[ \mu_\alpha(\check{\mathcal{E}})={-}\frac{1}{r}\sum_{ \check F\prec \check P} i_{\check F}(\det \check{\mathcal{E}})\,\mathrm{deg}_{\alpha}(D_{\check F}). \]

By the genericity assumption on $(G,\,\gamma )$, from lemma 3.6, we deduce a bijective correspondence $F \leftrightarrow \check F$ between facets of $P^{s}$ and facets of $\check P$. Thus

\[ \mu_\alpha(\check{\mathcal{E}})={-}\frac{1}{r}\sum_{ F\prec P^{s}} i_{\check F}(\det \check{\mathcal{E}})\,\mathrm{deg}_{\alpha}(D_{\check F}), \]

whereas by lemma 4.4,

\begin{align*} \mu_L(\mathfrak{P}_\mathbf{i}(\check{\mathcal{E}})) & = \mu_L(\mathcal{E}),\\ & ={-}\frac{1}{r}\sum_{F\prec P}i_F(\det \mathcal{E})\,\mathrm{deg}_L(D_F),\\ & ={-}\frac{1}{r}\sum_{F\prec P^{s}} i_{\check F}(\det \check{\mathcal{E}})b_F\mathrm{deg}_L(D_F) - \sum_{F\prec P^{us}}i_F\mathrm{deg}_L(D_F). \end{align*}

Consider the right-hand term of the last equality. We note two points:

1. (1) The second term $\sum _{F\prec P^{us}}i_F\mathrm {deg}_L(D_F)$ is independent of the reflexive sheaf $\check {\mathcal {E}}$. It depends only on the lattice volumes of the unstable facets $F\prec P^{us}$ and on the indices $i_F$ that determine the pull-back functors.

2. (2) The first term coincides with $\mu _\alpha (\check{\mathcal {E}})$ if for all $F\prec P^{s}$, $\mathrm {deg}_\alpha (D_{\check F})= b_F\mathrm {deg}_L(D_F)$, which, by proposition 4.9, is equivalent to $\mathrm {latvol(\check F)}=b_F\mathrm {deg}_L(D_F)$.

By proposition 4.11, we can chose the polytope $\check {P}\subseteq \check {M}_\mathbb {R}$ such that its facets satisfy $\mathrm {latvol}(\check {F})=b_F\mathrm {deg}_L(D_F)$ if and only if

\[ \sum_{\check F\prec \check P}b_F\mathrm{deg}_L(D_F)\check{u}_{ F}=0, \]

which, by lemma 3.8, is to say

\[ \sum_{F\prec P^{s}}\mathrm{deg}_L(D_F)u_F=0 \mod N_0\otimes_\mathbb{Z} \mathbb{R}. \]

Thus, the Minkowski condition implies the existence of the desired polytope and of $\alpha$.

Applying now (4.3) to the sheaf $\mathcal {O}(D_{\check F})$, using corollary 4.6, we deduce that $\mathrm {deg}_\alpha D_{\check F}=b_F \mathrm {deg}_L(D_F)$ for all $\check F \subset \check P$. Then, the uniqueness statement of theorem 4.7 follows from the equality $\mathrm {latvol}(\check F)=\mathrm {deg}_\alpha (D_{\check F})$ and unicity in proposition 4.11.

As noted above in proposition 4.2 (see [Reference Dasgupta, Dey and Khan4, Reference Hering, Nill and Suess9]), to test for stability, it is sufficient to consider equivariant reflexive subsheaves of the form $\check {\mathcal {E}}_W=\mathfrak {K}(\mathbb {E}\cap W)$ where $W$ runs through vector subspaces of $E$. The slopes then satisfy

\[ \mu_L(\mathcal{E})-\mu_L(\mathcal{E}_W)=\mu_\alpha(\check{\mathcal{E}})-\mu_\alpha(\check{\mathcal{E}}_W) \]

from which the equivalence of the stability conditions follows.

Proof of proposition 4.8. The ‘if’ statement is given in the previous theorem. We suppose that for some ample $\mathbb {R}$-divisor $\alpha$ on $Y$ the various stability notions are preserved by the pull-back functors. The ample divisor $\alpha$ defines a real polytope $\check P\subseteq \check {M}_\mathbb {R}= N_0^{\perp }\otimes _\mathbb {Z}\mathbb {R}$ whose facets $\check {F}$ have primitive normals $\check{u}_{F}=u_{F\cap U}\in N/N_0$.

We consider the functor $\mathfrak {P}_\mathbf {i}$ with $i_F=0$ for all unstable facets $F\prec P^{us}$. For any two $F_1,\,F_2\prec P^{s}$ consider the direct sum of rank one reflexive sheaves $\mathcal {O}_Y(d_1 D_{\check F_1})\oplus \mathcal {O}_Y(d_2 D_{\check F_2})$ where $d_1=b_{F_2}\mathrm {deg}_L(D_{F_2})$ and $d_2=b_{F_1}\mathrm {deg}_L(D_{F_1})$. Then, using lemma 4.5,

\[ \mathfrak{P}_\mathbf{i}(\mathcal{O}_Y(d_1 D_{\check F_1})\oplus \mathcal{O}_Y(d_2 D_{\check F_2})) =\mathcal{O}_X(d_1 b_{F_1} D_{F_1 })\oplus \mathcal{O}_X(d_2 b_{F_2}D_{F_2 }) \]

is the direct sum of rank-one reflexive sheaves of the same slope. By hypothesis, the initial sheaf must also be $\mu$-polystable so for any two stable facets of $P$,

\[ \mathrm{deg}_\alpha(b_{F_2}\mathrm{deg}_L(D_{F_2})D_{\check F_1}) = \mathrm{deg}_\alpha(b_{F_1}\mathrm{deg}_L(D_{F_1})D_{\check F_2}) \]

so the expression

\[ c=b_F\frac{\mathrm{deg}_L(D_F)}{\mathrm{deg}_\alpha(D_{\check F})} \]

is independent of $F\prec P^{s}$. Then, $\mathrm {latvol}(\check {F})=\mathrm {deg}_\alpha (D_{\check {F}})$ and by the lattice Minkowski theorem (proposition 4.11),

\[ \sum_{F\prec P^{s}} \mathrm{deg}_\alpha(D_{\check F}) \check{u}_{F}=0. \]

That is, by lemma 3.8,

\[ \frac{1}{c}\sum_{F\prec P^{s}} b_F\,\mathrm{deg}_L(D_F)\,b_F^{{-}1}\pi(u_F)=0 \]

as desired.

5.1 Compatible one parameter subgroups

Let $(X,\,L)$ be a polarized projective toric variety of complex dimension $n$ with torus $T=N\otimes _\mathbb {Z} \mathbb {C}^{*}$ and fan $\Sigma$. When considering subtori $G\subset T$, we will always assume that $G=N_0\otimes _\mathbb {Z} \mathbb {C}^{*}$, for a saturated sublattice $N_0$.

We will see that the Minkowski condition is very restrictive, in the sense that in most cases, for fixed $(X,\,L)$, there are few compatible one dimensional subtori. To see this, set $\mathrm {Div}_{\mathrm {ir}}^{T}(X)= \lbrace D_\rho,\, \: \rho \in \Sigma {(1)} \rbrace$ be the set of reduced and irreducible $T$-invariant divisors on $X$. Let $\mathbf {D}\subset \mathrm {Div}_{\mathrm {ir}}^{T}(X)$ be a non empty subset. Assume that there is a one parameter subgroup $G=N_0\otimes _\mathbb {Z} \mathbb {C}^{*}\subset T$ and a linearization $\gamma$ for $T$ on $L$ such that $(G,\,\gamma )$ is generic and the set of stable divisors for $(G,\,\gamma )$ is $\mathbf {D}$. Then, if Minkowski condition is satisfied, we have

\[ \sum_{D_\rho \in \mathbf{D}} \mathrm{deg}_L(D_\rho)\: u_\rho = 0 \mod N_0\otimes_\mathbb{Z} \mathbb{R} \]

Consider the element of $N$:

\[ u_{\mathbf{D}}:= \sum_{D_\rho\in \mathbf{D}} \mathrm{deg}_L(D_\rho)\: u_\rho. \]

Assume that $u_{\mathbf {D}}$ is not zero. Then Minkowski condition implies that

\[ N_0=N\cap(\mathbb{R}\cdot u_\mathbf{D}), \]

so that in particular $G$ is entirely determined by $u_\mathbf {D}$ and thus by the set $\mathbf {D}$. So for a given $\mathbf {D}$ with $u_\mathbf {D}$ non-zero, there is at most one compatible one-parameter subgroup of $T$ with $\mathbf {D}$ as set of stable divisors. As the number of stable divisors is the number of faces of a given polytope in $M_\mathbb {R}$ that intersects the hyperplane $N_0^{\perp }\otimes _\mathbb {Z} \mathbb {R}$, in the generic situation, we must have $\sharp \mathbf {D}\in \lbrace n,\, \ldots,\, \sharp \Sigma (1)-1 \rbrace$. Thus we have proved:

For example, the conditions of lemma 5.2 are satisfied by Hirzebruch surfaces $\mathbb {F}_a$ for $a\geq 2$, but not by $\mathbb {P}^{1}\times \mathbb {P}^{1}$.

Remark 5.3 Note that if $X$ is a projective toric orbifold, then from the sequence

\[ 0 \to M \to \bigoplus_{\rho\in \Sigma(1)} \mathbb{Z}\cdot D_\rho \to \mathrm{Cl}(X) \to 0 \]

and from the fact that the Picard group as finite index in the class group $\mathrm {Cl}(X)$, we deduce that $\sharp \Sigma (1)= n + p$, where $p$ is the Picard rank of $X$ (see [Reference Cox, Little and Schenck2, theorem 4.1.3 and proposition 4.2.7]). This gives the bound $\sum _{k=n}^{n+p - 1} \binom {n+p}{k}$ on the number of compatible one-parameter subgroups. In particular, for the complex projective space $\mathbb {C}\mathbb {P}^{n}$, this bound equals $n+1$. Using proposition 5.4, we see that for a given dimension, this bound is achieved.

5.2 Quotients to weighted projective spaces

In this section we will show:

From this proposition, we deduce that one can obtain $\mu$-stable reflexive sheaves on any polarized toric orbifold from $\mu$-stable reflexive sheaves on weighted projective spaces.

Proof. The argument is local at a vertex $v\in P$. As $X$ is an orbifold, its fan is simplicial, and thus the set $\lbrace u_F,\,\: v\in F\rbrace$ is a $\mathbb {Q}$-basis of $M_\mathbb {Q}$. Consider

\[ \mathbf{D}_v:=\lbrace D_F:\: v\in F \rbrace \]

and

\[ u_{\mathbf{D}_v}=\sum_{D_F\in \mathbf{D}_v} \mathrm{deg}_L(D_F) u_F=\sum_{v\in F} \mathrm{deg}_L(D_F) u_F. \]

As all the degrees $\mathrm {deg}_L(D_F)$ are positive, it is clear that $u_{\mathbf {D}_v}$ is not zero. Let $N_0=N\cap (\mathbb {R}\cdot u_{\mathbf {D}_v})$ and let $G=N_0\otimes _\mathbb {Z} \mathbb {C}^{*}$. Then, by construction of $u_{\mathbf {D}_v}$, up to dilation and translations of $P$, we can assure that $P$ intersects $N_0^{\perp }$ transversally and precisely along the faces $F$ that contains $v$. Thus, up to scaling $L$ and choosing an appropriate linearization, the set of stable divisors for the $G$ action is $\mathbf {D}_v$. By construction, Minkowski condition is satisfied and $G$ is compatible with $(X,\,L)$. The associated quotient is a weighted projective space as the number of stable facets is $n$, giving $n$ rays for the $(n-1)$-dimensional toric GIT quotient. To conclude, the vertices in the polytope correspond to $0$-dimensional orbits, hence to fixed points. Thus, there are at least $m$ such quotients, hence the result.

Remark 5.6 From lemma 5.2 and proposition 5.4, we deduce that the compatible one parameter subgroups for $(\mathbb {C}\mathbb {P}^{n},\,\mathcal {O}_{\mathbb {P}^{n}}(1))$ are given by the $\mathbb {C}^{*}$-actions

\[ \lambda \cdot [z_0,\ldots, z_n]=[z_0,\ldots, \lambda \, z_i, \ldots, z_n],\: i\in \lbrace 0, \ldots , n\rbrace, \]

and the associated quotients are all isomorphic to $\mathbb {C}\mathbb {P}^{n-1}$.

Example 5.7 Let $X=\mathbb {C}\mathbb {P}^{1}\times \mathbb {C}\mathbb {P}^{1}\times \mathbb {C}\mathbb {P}^{1}$ together with the polarization $L=\pi _1^{*}\mathcal {O}_{\mathbb {P}^{1}}(4)\otimes \pi _2^{*}\mathcal {O}_{\mathbb {P}^{1}}(4)\otimes \pi _3^{*}\mathcal {O}_{\mathbb {P}^{1}}(4)$ where $\pi _j: X\to \mathbb {C}\mathbb {P}^{1}$ denotes the projection onto the $j$-th factor. The associated polytope $P$ in $\mathbb {Z}^{3}\otimes _\mathbb {Z}\mathbb {R}=\mathbb {R}^{3}$ is a cube with edges of length $4$. Up to translation, that is up to changing the linearisation of the torus action on $L$, we can assume that $P$ is

\[ P=[{-}3,1]\times[{-}3,1]\times[{-}3,1]\subset\mathbb{R}^{3}. \]

Consider now the one parameter subgroup $G$ of $T$ whose lattice is $N_0=\mathbb {Z}\cdot (1,\,1,\,1)$. The GIT quotient of $(X,\,L)$ by $G$ is the toric variety $Y$ associated to the polytope $U\cap P=(1,\,1,\,1)^{\perp }\cap P$, that is

\[ P_Y=\lbrace (x,y,z)\in\mathbb{R}^{3}:\,: x+y+z=0,\: x,y,z\in[{-}3,1]\rbrace. \]

We recognize the polytope of $\mathbb {C}\mathbb {P}^{2}$, so that $Y\simeq \mathbb {C}\mathbb {P}^{2}$. The stable facets in $P^{s}$ are given by $F_x=\lbrace x=1 \rbrace \cap P$, $F_y=\lbrace y=1 \rbrace \cap P$ and $F_z=\lbrace z=1 \rbrace \cap P$. The degree of the associated invariant divisors $D_x,\, D_y$ and $D_z$ is equal to $16$ with respect to $L$, so the Minkowski condition becomes

\[ 16 u_{D_x} + 16 u_{D_y} + 16 u_{D_z}\in N_0\otimes_\mathbb{Z}\mathbb{R}=\mathbb{R}\cdot(1,1,1). \]

As $u_{D_x}=(-1,\,0,\,0)$, $u_{D_y}=(0,\,-1,\,0)$ and $u_{D_z}=(0,\,0,\,-1)$, the Minkowski condition is satisfied. By theorem 1.1, and from the fact that, up to scaling, the only polarization on $\mathbb {C}\mathbb {P}^{2}$ is $\mathcal {O}_{\mathbb {P}^{2}}(1)$, an equivariant reflexive sheaf $\check{\mathcal {E}}$ on $\mathbb {C}\mathbb {P}^{2}$ is slope stable with respect to $\mathcal {O}_{\mathbb {P}^{2}}(1)$ if and only if $\mathfrak {P}_\mathbf {i}(\check{\mathcal {E}})$ is slope stable on $X$ with respect to $L$. For example, the tangent sheaf $\mathcal {T}_{\mathbb {C}\mathbb {P}^{2}}$ is slope stable with respect to $\mathcal {O}_{\mathbb {P}^{2}}(1)$ (see e.g. [Reference Hering, Nill and Suess9, theorem 1.3]). Thus we obtain a slope stable rank $2$ equivariant reflexive sheaf on $(X,\,L)$ by setting $\mathcal {E}=\mathfrak {P}_{\bf 0}(\mathcal {T}_{\mathbb {C}\mathbb {P}^{2}})$. Note that this equivariant sheaf is not locally free, and hence $\mathfrak {P}_{\bf 0}$ doesn't preserve local freeness in general. To see this, recall from [Reference Klyachko12] that the family of filtrations defining $\mathcal {T}_{\mathbb {C}\mathbb {P}^{2}}$ is given by $\lbrace (T^{\check F}(i))\subset N_Y\otimes _\mathbb {Z} \mathbb {C}: \check F\prec P_Y,\, i\in \mathbb {Z}\rbrace$ where

\[ T^{\check F}(i)=\left\{\begin{array}{@{}ccc@{}} 0 & \: \mathrm{ if } \: & i\leq{-}2\\ \mathrm{Vect}(\check{u}_F) & \: \mathrm{ if } \: & i={-}1\\ N_Y\otimes_\mathbb{Z} \mathbb{C} & \: \mathrm{ if } \: & i\geq 0. \end{array} \right. \]

We compute $b_{F_x}=b_{F_y}=b_{F_z}=1$ here, so that the family of filtrations associated to $\mathcal {E}$ is $\mathbb {E}=\lbrace ( E^{F}(i))\subset E: F\prec P,\, i\in \mathbb {Z} \rbrace$, where in particular $E^{F}(i)=T^{F\cap U}(i)$ for $F\in \lbrace F_x,\, F_y,\, F_z \rbrace$. Klyachko's criterion for $\mathcal {E}$ to be locally free imposes a condition on the family of filtrations $\mathbb {E}$ for each face of $P$ (see e.g. [Reference Perling20, theorem 5.22] for the statement with increasing filtrations). In particular, for the vertex $(1,\,1,\,1)\in P$, this condition states that there should be a $\mathbb {Z}^{3}$-grading of $E=N_Y\otimes _\mathbb {Z} \mathbb {C}$:

\[ E=\bigoplus_{m\in \mathbb{Z}^{3}} E_m \]

such that for each face $F\prec P$ containing $(1,\,1,\,1)$, one has

\[ E^{F}(i)=\sum_{\langle m,u_F\rangle\leq i} E_m. \]

As for such faces, $\check{u}_F=\pi (u_F)$, where $\pi : \mathbb {Z}^{3} \to N_Y=\mathbb {Z}^{3}/\mathbb {Z}\cdot (1,\,1,\,1)$ denotes the projection map, and $E^{F}(i)=T^{U\cap F}(i)$, we deduce that we would have

\[ E_{(1,0,0)}=\mathrm{Vect}(\check{u}_{F_x}),\: E_{(0,1,0)}=\mathrm{Vect}(\check{u}_{F_y}),E_{(0,0,1)}=\mathrm{Vect}(\check{u}_{F_z}). \]

But then $\dim (E)\geq 3$, which is absurd as $\dim (E)=\mathrm {rk}(N_Y)=2$. Hence $\mathbb {E}$ doesn't satisfy Klyachko's criterion and $\mathcal {E}$ is not locally free.

5.3 Projectivization of toric vector bundles

As a motivating example, we consider the variety given as the projectivization of a torus invariant vector bundle. We refer to [Reference Lazarsfeld15, appendix A] and [Reference Cox, Little and Schenck2, section 7.3] for the construction and classical results. Let $Y$ be a projective toric variety with fan $\Sigma _Y$. For $i=1,\,\ldots,\,r$, let $D_i=\sum _{\rho \in \Sigma _Y(1)} a_{i\rho }D_\rho$ be invariant divisors on $Y$ and let $V_\mathcal {F}$ be the vector bundle associated to the locally free sheaf

\[ \mathcal{F}=\mathcal{O}_Y\oplus\mathcal{O}_Y(D_1)\oplus\cdots\oplus\mathcal{O}_Y(D_r). \]

Let $X=\mathbb {P}(V_\mathcal {F}^{\vee })$ be the projectivization of the dual of $V_\mathcal {F}$, with $\pi :X\to Y$ the projection to the base $Y$. Then $X$ is a projective toric variety, with torus $T_X=T_Y\times (\mathbb {C}^{*})^{r}$. Denote by $N_X,\, N_Y$ the lattices of $X$ and $Y$, and by $M_X$ and $M_Y$ their duals. Then $N_X=N_Y\times \mathbb {Z}^{r}$ and we have exact sequences

\begin{align*} & 0\longrightarrow \mathbb{Z}^{r}\longrightarrow N_Y\times \mathbb{Z}^{r}\longrightarrow N_Y \longrightarrow 0,\\ & 0\longrightarrow M_Y\longrightarrow M_Y\times(\mathbb{Z}^{r})^{*}\longrightarrow (\mathbb{Z}^{r})^{*}\longrightarrow 0. \end{align*}

Let $L_Y$ be an ample line bundle on $Y$ and let $\mathcal {O}_X(1)=\mathcal {O}_{\mathbb {P}(V_\mathcal {F})}(1)$ be the Serre line bundle on $X$. Then, up to replacing $L_Y$ by a sufficiently high power, $L_X=\pi ^{*}L_Y\otimes \mathcal {O}_X(1)$ is an ample line bundle on $X$. Assume now that $L_Y=\mathcal {O}(D_Y)$ for an ample divisor on $Y$

\[ D_Y=\sum_{\rho\in\Sigma_Y(1)} b_\rho D_\rho, \]

with associated lattice polytope

\[ P_Y=\{m\in (M_Y)_\mathbb{R}\,:\, \langle m,u_\rho\rangle\geq{-}b_\rho\: \textrm{ for all } \rho\in\Sigma_Y(1)\}. \]

To determine a polytope $P_X\subseteq (M_Y)_\mathbb {R}\times \mathbb {R}^{r}$ associated to $(X,\,L_X)$, we first describe the invariant irreducible divisors of $X$. They are of two types:

\[ \mathrm{Div}_{irr}^{T}(X)=\lbrace \pi^{{-}1}(D_\rho): \rho\in \Sigma_Y(1)\rbrace \cup \lbrace\{s_i =0\}\textrm{ for } i=0,1,\ldots,r \rbrace \]

where the $\lbrace s_i = 0 \rbrace$ are the relative hyperplane sections associated to the line subbundles of $V_\mathcal {E}^{\vee }$. Set $\hat D_\rho = \pi ^{-1}(D_\rho )$ for all $\rho \in \Sigma _Y(1)$ and $D_{s_i}:=\lbrace s_i = 0 \rbrace$ for all $i\in \lbrace 0,\, \ldots,\, r\rbrace$. Then $L_X=\mathcal {O}(\hat D)$ with

\[ \hat D = \sum_{\rho\in\Sigma_Y(1)} b_\rho \hat D_\rho + \sum_{i=0}^{r} D_{s_i} \]

and associated polytope $P_X$ defined by $(m_y,\,m_x)\in (M_Y)_\mathbb {R}\times (\mathbb {R}^{r})^{*}$ is in $P_X$ if and only if

\begin{align*} & \langle m_x,e_i\rangle \geq{-}1, \text{ for all }i=0,\ldots,r, \textrm{ and }\\ & \langle (m_y,m_x), u_\rho +\sum_{i=1}^{r} a_{i\rho}e_i\rangle \geq{-}b_\rho, \text{ for all }\rho\in\Sigma_Y(1), \end{align*}

where $\{e_i:i=1,\,\ldots,\, r\}$ is the standard basis of $\mathbb {R}^{r}$ and $e_0=-(e_1+\ldots + e_r)$.

We will perform a GIT quotient of $X$ by the torus $G=N_0\otimes _\mathbb {Z} \mathbb {C}^{*}$, where $N_0=\lbrace 0_Y \rbrace \times \mathbb {Z}^{r}$. Consider the linearization $\gamma _y$ for $T_Y$ on $L_Y$ giving the polytope $P_Y$ (recall § 3.1, equation (3.1)). Consider the standard linearization $\gamma _x$ of $(\mathbb {C}^{*})^{r}$ on $\mathcal {O}_{\mathbb {C}\mathbb {P}^{r}}(1)$ associated to the polytope

\[ \lbrace \langle m_x,e_i\rangle \geq{-}1, \text{ for all }i=0,\ldots,r \rbrace \subset (\mathbb{R}^{r})^{*}. \]

Combining the two gives the linearization $\gamma = \gamma _y \times \gamma _x$ of $T_X$ on $L_X$ associated to $P_X$. Then, setting $U=N_0^{\perp }\otimes _\mathbb {Z}\mathbb {R}$, we have $U=(M_Y)_\mathbb {R} \times \lbrace 0 \rbrace$. In particular, the only facets of $P_X$ that intersect $U$ are those given by equations $\langle m_y,\,u_\rho \rangle +\sum _i a_{i\rho }\langle m_x,\,e_i\rangle =-b_\rho$ for some $\rho \in \Sigma _Y(1)$ and the polytope $P_X \cap U$ is $P_Y\times \lbrace 0 \rbrace$. That is, the GIT quotient of $(X,\,L_X)$ by $G$ is $(Y,\,L_Y)$. We also note that $(G,\,\gamma )$ is generic.