---
abstract: |
    This paper proves that every projective toric variety is the fine moduli space for stable representations of an appropriate bound quiver.  To accomplish this, we study the quiver $Q$ with relations $R$ corresponding to the finite-dimensional algebra $\operatorname{End}\bigl(
      \textstyle\bigoplus\nolimits_{i=0}^{r} L_i \bigr)$ where $\mathcal{L} := (\mathscr{O}_X,L_1, \dotsc, L_r)$ is a list of line bundles on a projective toric variety $X$. The quiver $Q$ defines a smooth projective toric variety, called the multilinear series $|\mathcal{L}|$, and a map $X \rTo |\mathcal{L}|$. We provide necessary and sufficient conditions for the induced map to be a closed embedding.  As a consequence, we obtain a new geometric quotient construction of projective toric varieties.  Under slightly stronger hypotheses on $\mathcal{L}$, the closed embedding identifies $X$ with the fine moduli space of stable representations for the bound quiver $(Q,R)$.
address:
- |
    Department of Mathematics\
    University of Glasgow\
    University Gardens\
    Glasgow\
    G12 8QW\
    United Kingdom
- |
    Department of Mathematics and Statistics\
    Queen’s University\
    Kingston\
    ON\
    K7L 3N6\
    Canada
author:
- Alastair Craw
- 'Gregory G. Smith'
title: |
    Projective Toric Varieties as\
    fine moduli spaces of Quiver Representations
---

Introduction {#sec:introduction}
============

The dictionary between the geometry of a moduli space $X$ and the family of objects classified by $X$ lies at the heart of modern algebraic geometry.  Fine moduli spaces, although substantially rarer than coarse ones, provide the fundamental example of this correspondence.  A scheme $X$ is a fine moduli space for the equivalence classes of some objects if and only if there is a universal family of the selected objects over $X$ such that every other family of objects is induced from the universal one by a unique morphism to $X$. The hyperplane bundle on ${\ensuremath{\mathbb{P}}}^d$ and the tautological vector bundle on a Grassmannian are the classic examples of universal families.  A universal family is a powerful tool for studying the geometry of $X$ as illustrated by the forgetful morphism between moduli spaces of pointed stable curves, or the Fourier-Mukai transform on abelian varieties.  The primary goal of this paper is to realize every projective toric variety as a fine moduli space of stable representations for an appropriate bound quiver. 

To be more precise, let $X$ be a projective toric variety over a field ${\ensuremath{\Bbbk}}$ and consider a list $\mathcal{L} := (\mathscr{O}_X, L_1, \dotsc,
L_r)$ of line bundles on $X$; for convenience set $L_0 :=
\mathscr{O}_X$. The finite-dimensional ${\ensuremath{\Bbbk}}$-algebra $\operatorname{End}\bigl(
\textstyle\bigoplus\nolimits_{i=0}^{r} L_i \bigr)$ is encoded by a quiver $Q$, called the complete quiver of sections for $\mathcal{L}$, together with an ideal of relations $R$ in the path algebra ${\ensuremath{\Bbbk}}Q$. We associate to the quiver $Q$ a unimodular, projective toric variety $|\mathcal{L}|$ called the multilinear series of $Q$. The variety $|\mathcal{L}|$ can be defined combinatorially, by geometric invariant theory, or via representation theory; see Proposition \[pro:smooth\].  Since the isomorphism ${\ensuremath{\Bbbk}}Q/R \cong
\operatorname{End}\bigl( \bigoplus_{i=0}^{r} L_i \bigr)$ identifies arrows in $Q$ with global sections of line bundles, the quiver $Q$ induces a map from $X$ to $|\mathcal{L}|$. We prove the following:

\[thm:1\] If $L_1, \dotsc, L_r$ are basepoint-free line bundles on $X$, then the induced map $\varphi_{|\mathcal{L}|} \colon X \rTo |\mathcal{L}|$ is a morphism and the image is presented as a geometric quotient. 

When $r = 1$, $\varphi_{|\mathcal{L}|}$ coincides with the morphism from $X$ to the linear series $|L_1|$.

Lists $\mathcal{L}$ for which the induced morphism $\varphi_{|
  \mathcal{L}|}$ is a closed embedding are ubiquitous; see Proposition \[pro:veryample\].  Hence, Theorem \[thm:1\] produces a wealth of new geometric quotient constructions for a projective toric variety $X$. In particular, these *geometric* quotients provide new “homogeneous coordinates” for the points on $X$. In contrast with [@Cox; @Kajiwara], these quotient constructions are not intrinsic to the toric variety; they depend on the choice of line bundles in $\mathcal{L}$. Although we recover the quotient constructions in [@Cox; @Kajiwara] for some toric varieties and particular lists $\mathcal{L}$, the homogeneous coordinate systems arising from a multilinear series $|\mathcal{L}|$ are typically larger.  Examples suggest that some of these larger coordinate systems appear naturally in the quantum cohomology of $X$ and the derived category of coherent sheaves on $X$.

To achieve the primary goal, we relate the image $\varphi_{|\mathcal{L}|}(X)$ to an important subscheme of $|\mathcal{L}|$. From the viewpoint of representation theory, the multilinear series $|\mathcal{L}|$ is the fine moduli space of $\vartheta$-stable representations with dimension vector $(1, \dotsc,
1)$ for the quiver $Q$ where $\vartheta$ is a distinguished weight on $Q$; see Proposition \[pro:smooth\].  The ideal of relations $R$ in the path algebra ${\ensuremath{\Bbbk}}Q$ determines a subscheme of $|\mathcal{L}|$ that coincides with the fine moduli space $\mathcal{M}_\vartheta(Q,R)$ of $\vartheta$-stable representations with dimension vector $(1,
\dotsc, 1)$ for the bound quiver $(Q,R)$. In other words, the variety $\mathcal{M}_\vartheta(Q,R)$ classifies certain finite-dimensional modules over the ${\ensuremath{\Bbbk}}$-algebra $\operatorname{End}\bigl( \bigoplus_{i=0}^{r} L_i
\bigr)$. The universal $\vartheta$-stable representation of $(Q,R)$ over $\mathcal{M}_\vartheta(Q,R)$ decomposes into a direct sum of line bundles called the tautological line bundles.  Our main results can be summarized as follows:

\[thm:2\] Let $X$ be a projective toric variety.  There exist (many) lists $\mathcal{L}$ of line bundles on $X$ such that the induced morphism $\varphi_{|\mathcal{L}|} \colon X \rTo |\mathcal{L}|$ identifies $X$ with the fine moduli space $\mathcal{M}_\vartheta(Q,R)$. Moreover, the tautological line bundles on $\mathcal{M}_\vartheta(Q,R)$ coincide with the line bundles in the list $\mathcal{L}$.

This fine moduli interpretation yields a functorial approach to projective toric varieties.  Specifically, it allows one to describe the data needed to specify a map from a scheme to a projective toric variety as in [@Cox2; @Kajiwara]. 

Theorem \[thm:2\] also helps clarify the relationship between descriptions of the derived category $D^{b}(\mathcal{O}_X\operatorname{-mod})$ and realizations of $X$ as a fine moduli space of quiver representations. [ @Bondal] shows that $D^{b}(\mathcal{O}_X\operatorname{-mod})$ is equivalent to the derived category of finite-dimensional modules over $\operatorname{End}\bigl( \bigoplus_{i=0}^{r}
\mathscr{F}_i \bigr)$ if and only if the coherent sheaves $\mathscr{F}_{i}$ form a complete strong exceptional collection on $X$. On certain toric quiver varieties, [@AH] describe such collections in which the $\mathscr{F}_i$ are line bundles; toric quiver varieties are fine moduli spaces of quiver representations.  The influential [@King2] constructs complete strong exceptional collections of line bundles on several smooth toric surfaces by realizing the surfaces as fine moduli spaces of stable representations of a bound quiver.  Given a smooth projective variety with a complete strong exceptional collection of line bundles, [@BergmanProudfoot] establishes that the variety is isomorphic to a connected component of a corresponding moduli space of stable quiver representations. [ @Kawamata] proves that every toric variety has a complete exceptional collection of coherent sheaves and, in contrast, [@HillePerling] exhibits a smooth toric surface that does not have a complete strong exceptional collection of line bundles.  In this context, Theorem \[thm:2\] clearly differentiates between a fine moduli interpretation of a variety and the existence of a complete strong exceptional collection of line bundles. 

This paper is organized as follows.  Our notation and some standard results from toric geometry and quiver theory are described in §\[sec:background\].  In §\[sec:quiver\], we define a quiver of sections and its associated multilinear series.  This generalizes the classical notion of a linear series from a single line bundle to a list of line bundles.  The induced map to the multilinear series is studied in §\[sec:multilinear\].  In particular, we give necessary and sufficient conditions for the induced map to be a morphism or a closed embedding.  Finally, §\[sec:boundreps\] examines representations of a bound quiver of sections and establishes our main results. 

We thank Diane Maclagan, Sorin Popescu, Nicholas Proudfoot and Karen Smith for stimulating conversations.  We also thank an anonymous referee for valuable comments and suggestions.  This paper owes much to experiments made with [@M2].  The second author is partially supported by NSERC. 

Background and Notation {#sec:background}
=======================

We collect here standard definitions, results and notation.  In this paper, ${\ensuremath{\mathbb{N}}}$ denotes the nonnegative integers and ${\ensuremath{\Bbbk}}$ is an algebraically closed field of characteristic zero. 

Toric Varieties {#sub:toric}
---------------

Let $X$ be a projective toric variety over ${\ensuremath{\Bbbk}}$ determined by a strongly convex rational polyhedral fan $\Sigma_{X} \subseteq N_X
\otimes_{{\ensuremath{\mathbb{Z}}}} {\ensuremath{\mathbb{R}}}\cong {\ensuremath{\mathbb{R}}}^{d}$ where $N_X$ is a lattice of rank $d$. The dual lattice is $M_X := \operatorname{Hom}_{{\ensuremath{\mathbb{Z}}}}(N_X,{\ensuremath{\mathbb{Z}}})$ and $T_X := N_X
\otimes_{{\ensuremath{\mathbb{Z}}}} {\ensuremath{\Bbbk}}^{*}$ is the algebraic torus acting on $X$. The $i$-dimensional cones of $\Sigma_{X}$ form the set $\Sigma_{X}(i)$. Since $X$ is projective, $\bigcup_{\sigma \in \Sigma_{X}} \sigma =
N_{X} \otimes_{{\ensuremath{\mathbb{Z}}}} {\ensuremath{\mathbb{R}}}$ and $\Sigma_{X}(d)$ is the set of maximal cones. 

Each $\rho \in \Sigma_{X}(1)$ corresponds to an irreducible $T_X$-invariant Weil divisor $D_{\rho}$ on $X$. These divisors generate the free abelian group ${\ensuremath{\mathbb{Z}}}^{\Sigma_{X}(1)}$ of $T_X$-invariant Weil divisors and the semigroup ${\ensuremath{\mathbb{N}}}^{\Sigma_{X}(1)}$ of effective $T_X$-invariant Weil divisors.  The quotient of ${\ensuremath{\mathbb{Z}}}^{\Sigma_{X}(1)}$ by the subgroup of principal divisors is the class group (or Chow group) $\operatorname{Cl}(X)$. The $T_X$-invariant Cartier divisors $\operatorname{CDiv}(X)$ form a subgroup of ${\ensuremath{\mathbb{Z}}}^{\Sigma_{X}(1)}$. Moreover, there is a commutative diagram $$\label{eq:fundiagram}
  \begin{diagram}[size=2em]
    0 & \rTo & M_X & \rTo & \operatorname{CDiv}(X) & \rTo & \operatorname{Pic}(X) & \rTo
    & 0 \\
    & & \dEqual & & \dTo & & \dTo & & \\
    0 & \rTo & M_X & \rTo & {\ensuremath{\mathbb{Z}}}^{\Sigma_{X}(1)} & \rTo & \operatorname{Cl}(X) & \rTo
    & 0
  \end{diagram}$$ where the rows are exact and the vertical arrows are inclusions; see §3.4 in [@Fulton].  The projection from ${\ensuremath{\mathbb{Z}}}^{\Sigma_{X}(1)}$ to $\operatorname{Cl}(X)$ is denoted by $u \mapsto [u]$ and the inclusion of $\operatorname{Pic}(X)$ into $\operatorname{Cl}(X)$ is also denoted by $L \mapsto [L]$. For a line bundle $L$ on $X$ and a global section $s \in H^0(X,L)$, ${\operatorname{div}}(s)$ denotes the effective Cartier divisor determined by $s$.

The total coordinate ring of $X$ is the polynomial ring $S_X :=
{\ensuremath{\Bbbk}}[x_{\rho} : \rho \in \Sigma_{X}(1)]$. Following [@Cox], we regard $S_X$ as the semigroup algebra of ${\ensuremath{\mathbb{N}}}^{\Sigma_{X}(1)}$ with the $\operatorname{Cl}(X)$-grading induced by $\deg(x^u) = \deg\bigl(
\prod_{\rho \in \Sigma_{X}(1)} x_{\rho}^{u_{\rho}} \bigr) = [u] \in
\operatorname{Cl}(X)$. A divisor $D = \sum_{\rho \in \Sigma_{X}(1)}
u_{\rho}D_{\rho}$ determines a Laurent monomial $x^u = \prod_{\rho \in
  \Sigma_{X}(1)} x_{\rho}^{u_{\rho}} \in {\ensuremath{\Bbbk}}[x_{\rho}^{\pm 1} : \rho
\in \Sigma_{X}(1)]$ and we often write the monomial as $x^D$. The support of $D$ or $x^u$ is the set $$\operatorname{supp}(D) = \operatorname{supp}(x^u) = \{ \rho \in \Sigma_{X}(1) : u_{\rho} \neq 0 \}
\, .$$ For a cone $\sigma \in \Sigma_{X}$, $\widehat{\sigma}$ is the set of one-dimensional cones in $\Sigma_{X}$ that are not contained in $\sigma$ and $x^{\widehat{\sigma}} = \prod_{\rho \in \widehat{\sigma}}
x_{\rho}$ is the associated monomial in $S_X$. The irrelevant ideal of $X$ is the square-free (i.e. reduced) monomial ideal $B_{X} := (
x^{\widehat{\sigma}} : \sigma \in \Sigma_{X})$. Theorem 2.1 in [@Cox] shows that the pair $(S_X,B_X)$ encodes a quotient construction of $X$. Specifically, if ${\ensuremath{\mathbb{V}}}(B_X)$ is the subvariety of ${\ensuremath{\mathbb{A}}}^{\Sigma_X(1)}$ defined by $B_X$, then the toric variety $X$ is a categorical quotient of ${\ensuremath{\mathbb{A}}}^{\Sigma_{X}(1)} \setminus {\ensuremath{\mathbb{V}}}(B_{X})$ by the group $\operatorname{Hom}_{{\ensuremath{\mathbb{Z}}}}(\operatorname{Cl}(X), {\ensuremath{\Bbbk}}^{*})$; the group action is induced by the $\operatorname{Cl}(X)$-grading of $S_X$.

Quivers {#sub:quivers}
-------

A quiver $Q$ is specified by two finite sets $Q_0$ and $Q_1$, whose elements are called vertices and arrows, together with two maps $\operatorname{hd}, \operatorname{tl}\colon Q_1 \rTo Q_0$ indicating the vertices at the head and tail of each arrow.  A nontrivial path in $Q$ is a sequence of arrows $p = a_1 \dotsb a_\ell$ with $\operatorname{hd}(a_{k}) = \operatorname{tl}(a_{k+1})$ for $1 \leq k < \ell$. We set $\operatorname{tl}(p) = \operatorname{tl}(a_{1})$ and $\operatorname{hd}(p)
= \operatorname{hd}(a_\ell)$. Each $i \in Q_0$ gives a trivial path $e_i$ where $\operatorname{tl}(e_i) = \operatorname{hd}(e_i) = i$. The path algebra ${\ensuremath{\Bbbk}}Q$ is the ${\ensuremath{\Bbbk}}$-algebra whose underlying ${\ensuremath{\Bbbk}}$-vector space has a basis consisting of paths in $Q$; the product of two basis elements equals the basis element defined by concatenation of the paths if possible or zero otherwise.  A cycle is a path $p$ in which $\operatorname{tl}(p) = \operatorname{hd}(p)$. A quiver is acyclic if it contains no cycles.  A vertex is a source if it is not the head of any arrow and a quiver is rooted if it has a unique source. 

A walk $\gamma$ in $Q$ is an alternating sequence $i_0a_1i_1 \dotsb
a_\ell i_{\ell}$ of vertices $i_1, \dotsc, i_\ell$ and arrows $a_1,\dotsc, a_\ell$ where $a_{k}$ is an arrow between $i_{k-1}$ and $i_k$. If $\operatorname{tl}(a_k) = i_{k-1}$ and $\operatorname{hd}(a_k) = i_k$ then $a_k$ is a forward arrow in $\gamma$; otherwise $\operatorname{tl}(a_k) = i_k$, $\operatorname{hd}(a_k)
= i_{k-1}$ and $a_k$ is a backward arrow.  If $a \in Q_1$ then $a^{-1}$ denotes the walk from $\operatorname{hd}(a)$ to $\operatorname{tl}(a)$. A walk $\gamma$ is closed if $i_0 = i_\ell$ and a circuit is a closed walk in which the arrows $a_1, \dotsc, a_\ell$ are distinct.  A quiver is connected if there is a walk between any two vertices.  A tree is a connected acyclic quiver.  We say that $Q' \subseteq Q$ is a spanning subquiver if $Q_0' = Q_0$.

The vertex space ${\ensuremath{\mathbb{Z}}}^{Q_0}$ is the free abelian group of functions from $Q_0$ to ${\ensuremath{\mathbb{Z}}}$ and the arrow space ${\ensuremath{\mathbb{Z}}}^{Q_1}$ is the free abelian group of functions from $Q_1$ to ${\ensuremath{\mathbb{Z}}}$. The characteristic functions $\chi_{i} \colon Q_0 \rTo {\ensuremath{\mathbb{Z}}}$ for $i \in Q_0$ and $\chi_{a}
\colon Q_1 \rTo {\ensuremath{\mathbb{Z}}}$ for $a \in Q_1$ form the standard bases for the vertex and arrows spaces.  We write ${\ensuremath{\mathbb{N}}}^{Q_0}$ and ${\ensuremath{\mathbb{N}}}^{Q_1}$ for the semigroups generated by all ${\ensuremath{\mathbb{N}}}$-linear combinations of the characteristic functions $\chi_{i}$ and $\chi_{a}$ respectively.  The incidence map $\operatorname{inc}\colon {\ensuremath{\mathbb{Z}}}^{Q_1} \rTo {\ensuremath{\mathbb{Z}}}^{Q_0}$ is defined by $\operatorname{inc}(\chi_{a}) = \chi_{\operatorname{hd}(a)} - \chi_{\operatorname{tl}(a)}$. A function $\theta \colon Q_0 \rTo {\ensuremath{\mathbb{Z}}}$ is an integral weight of $Q$ if $\sum_{i
  \in Q_0} \theta_i = 0$ and a function $f \colon Q_1 \rTo {\ensuremath{\mathbb{Z}}}$ is an integral circulation if $$\sum\limits_{\substack{a \in Q_1\\ \operatorname{tl}(a) = i}} f_a =
\sum\limits_{\substack{a \in Q_1 \\ \operatorname{hd}(a) = i}} f_a \;\; \text{for
  each $i \in Q_0$.}$$ The weight lattice $\operatorname{Wt}(Q) \subset {\ensuremath{\mathbb{Z}}}^{Q_0}$ and the circulation lattice $\operatorname{Cir}(Q) \subset {\ensuremath{\mathbb{Z}}}^{Q_1}$ are generated by the integral weights and circulations respectively.  There is an exact sequence $$\label{eq:graphses}
  \begin{diagram}[size=2.5em,l>=2.5em,midshaft,labelstyle=\scriptstyle]
    0 & \rTo & \operatorname{Cir}(Q) & \rTo & {\ensuremath{\mathbb{Z}}}^{Q_1} & \rTo^{\operatorname{inc}} & \operatorname{Wt}(Q) 
  \end{diagram}$$ and the incidence map is surjective when $Q$ is connected; see §4 in [@Biggs].  For $a \in Q_1$ and a walk $\gamma$, let $\operatorname{mult}_{\gamma}(a) \in {\ensuremath{\mathbb{Z}}}$ equal the number of times $a$ appears as a forward arrow in $\gamma$ minus the number of times it appears as a backward arrow.  Given a walk $\gamma$, we set $f(\gamma) := \sum_{a
  \in Q_1} \operatorname{mult}_{\gamma}(a) \chi_{a} \in {\ensuremath{\mathbb{Z}}}^{Q_1}$; $f(\gamma) \in
\operatorname{Cir}(Q)$ if and only if $\gamma$ is a closed walk. 

Representations of Quivers {#sub:reps}
--------------------------

Let $Q$ be a connected quiver.  A representation $W = (W_i, w_a)$ of $Q$ consists of a ${\ensuremath{\Bbbk}}$-vector space $W_i$ for each $i \in Q_0$ and a ${\ensuremath{\Bbbk}}$-linear map $w_a \colon W_{\operatorname{tl}(a)} \rTo W_{\operatorname{hd}(a)}$ for each $a \in Q_1$. The dimension vector of $W$ is $\sum_{i\in Q_0}
\dim_{{\ensuremath{\Bbbk}}}(W_{i})\chi_i \in {\ensuremath{\mathbb{N}}}^{Q_0}$. In this paper, we will assume that $\dim_{{\ensuremath{\Bbbk}}}( W_{i}) = 1$ for all $i \in Q_0$. A map between representations $W = (W_i, w_a)$ and $W' = (W_i', w_a')$ is a family $\psi_{i} \colon W_i^{\,} \rTo W_i'$ for $i \in Q_0$ of ${\ensuremath{\Bbbk}}$-linear maps that are compatible with the structure maps, that is $w_a'
\psi_{\operatorname{tl}(a)} = \psi_{\operatorname{hd}(a)} w_a$ for all $a \in Q_1$. With composition defined componentwise, we obtain the abelian category of representations of $Q$. Each rational weight $\theta \in
\operatorname{Wt}(Q)\otimes_{{\ensuremath{\mathbb{Z}}}}{\ensuremath{\mathbb{Q}}}$ defines a stability notion for representations and subquivers of $Q$. A representation $W$ is $\theta$-stable if, for every proper, nonzero subrepresentation $W' \subset W$, we have $\theta(W') := \sum_{i \in \operatorname{supp}(W')} \theta_i > 0$, where $\operatorname{supp}(W')
:= \{ i \in Q_0 : W_i' \neq 0 \}$. The notion of $\theta$-semistability is obtained by replacing $>$ with $\geq$. A subquiver $Q' \subseteq Q$ is $\theta$-stable if it admits a $\theta$-stable representation. 

The isomorphism classes of representations are orbits in the representation space $${\ensuremath{\mathbb{A}}}^{Q_1} = \operatorname{Spec}\bigl( {\ensuremath{\Bbbk}}[y_a : a\in Q_1] \bigr) \cong
\bigoplus\limits_{a \in Q_1} \operatorname{Hom}_{{\ensuremath{\Bbbk}}}(W_{\operatorname{tl}(a)}, W_{\operatorname{hd}(a)})$$ by the action of the group $({\ensuremath{\Bbbk}}^*)^{Q_0} \cong \prod_{i \in Q_0}
\operatorname{GL}(W_i)$ induced by the incidence map; in other words, it acts by $(g \cdot w)_{a} = g_{\operatorname{hd}(a)}^{\,} w_{a}
g_{\operatorname{tl}(a)}^{-1}$. Hence, we have a faithful action of the algebraic torus $G := \operatorname{Hom}_{{\ensuremath{\mathbb{Z}}}} \bigl( \operatorname{Wt}(Q),{\ensuremath{\Bbbk}}^* \bigr)$ on ${\ensuremath{\mathbb{A}}}^{Q_1}$ and it gives a $\operatorname{Wt}(Q)$-grading of the polynomial ring $S_Y := {\ensuremath{\Bbbk}}[y_a :
a\in Q_1]$. For $\theta \in \operatorname{Wt}(Q)$, let $(S_Y)_{\theta}$ be the $\theta$-graded piece.  Following [@King1], the GIT-quotient $$\mathcal{M}_\theta(Q) := {\ensuremath{\mathbb{A}}}^{Q_1}{\ensuremath{/\!\!/_{\!\theta}\,}}G = \operatorname{Proj}\Biggl(
\bigoplus\limits_{k \in {\ensuremath{\mathbb{N}}}} (S_Y)_{k\theta} \Biggr)$$ is the categorical quotient $({\ensuremath{\mathbb{A}}}^{Q_1})^{\text{ss}}_\theta / G$, where $({\ensuremath{\mathbb{A}}}^{Q_1})^{\text{ss}}_\theta \subseteq {\ensuremath{\mathbb{A}}}^{Q_1}$ is the open subscheme parametrizing $\theta$-semistable representations of $Q$. Since $\dim(W_i) = 1$ for all $i \in Q_0$, $\mathcal{M}_\theta(Q)$ is also a toric quiver variety as defined in [@Hille1].  A weight $\theta\in \operatorname{Wt}(Q)\otimes_{{\ensuremath{\mathbb{Z}}}}{\ensuremath{\mathbb{Q}}}$ is generic if every $\theta$-semistable representation is $\theta$-stable.  In this case, $\mathcal{M}_\theta(Q)$ is the geometric quotient $({\ensuremath{\mathbb{A}}}^{Q_1})^{\text{s}}_\theta / G$, where $({\ensuremath{\mathbb{A}}}^{Q_1})^{\text{s}}_\theta$ parametrizes $\theta$-stable representations of $Q$. The set of generic weights decomposes into finitely many open chambers, where $\mathcal{M}_\theta(Q)$ is unchanged as $\theta$ varies in a chamber; see [@DolgachevHu; @Thaddeus2]. 

For generic $\theta$, Proposition 5.3 in [@King1] implies that $\mathcal{M}_\theta(Q)$ is the fine moduli space of $\theta$-stable representations of $Q$. To describe the universal family on $\mathcal{M}_\theta(Q)$, we set $Q_0 = \{ 0,
\dotsc, r \}$ and identify the group $G$ with $\{ (g_0, \dotsc, g_r)
\in ( {\ensuremath{\Bbbk}}^* )^{Q_0} : g_0 = 1 \}$. This choice determines a $G$-equivariant vector bundle $\bigoplus_{i \in Q_0}
\mathscr{O}_{{\ensuremath{\mathbb{A}}}^{Q_1}}$ which descends to $\bigoplus_{i \in Q_0} F_i$ on $\mathcal{M}_\theta(Q)$; see Proposition 5.3 in [@King1].  The line bundles $F_0, \dotsc, F_r$ are called the tautological line bundles on $\mathcal{M}_\theta(Q)$. Since $G$ acts trivially on the $0$-th component of $\bigoplus_{i \in Q_0} \mathscr{O}_{{\ensuremath{\mathbb{A}}}^{Q_1}}$, it follows that $F_0$ is the trivial line bundle. 

Quivers of Sections {#sec:quiver}
===================

The goal of this section is to extend the classical notion of a linear series from a single line bundle to a list of line bundles.  Let $\mathcal{L} := (L_0, \dotsc, L_r)$ be a list of distinct line bundles on the projective toric variety $X$. A $T_X$-invariant section $s \in
H^0(X,L_j^{\;} \otimes L_{i}^{-1})$ is *indecomposable* if the divisor ${\operatorname{div}}(s)$ cannot be expressed as a sum ${\operatorname{div}}(s') + {\operatorname{div}}(s'')$ where $s' \in H^0(X,L_k^{\;} \otimes L_{i}^{-1})$ and $s'' \in
H^0(X,L_j^{\;} \otimes L_{k}^{-1})$ are nonzero $T_X$-invariant sections and $0 \leq k \leq r$. A *quiver of sections* associated to $\mathcal{L}$ is a quiver $Q$ in which the vertices $Q_0
= \{ 0, \dotsc, r \}$ correspond to the line bundles in $\mathcal{L}$ and the arrows from $i$ to $j$ correspond to a subset of the indecomposable $T_X$-invariant sections in $H^0(X,L_j^{\;} \otimes
L_{i}^{-1})$. If every indecomposable $T_X$-invariant section in $H^0(X,L_j^{\;} \otimes L_{i}^{-1})$ for $0 \leq i, j \leq r$ corresponds to an arrow then $Q$ is the *complete quiver of sections* for $\mathcal{L}$. Since $X$ is projective, the unique element in $H^0(X,L_i^{\;} \otimes L_{i}^{-1}) = H^0(X,
\mathscr{O}_{X})$ defines the trivial path $e_i$ in $Q$. Moreover, if $L_j \neq L_i$ then projectivity implies that both $H^{0}(X,L_j^{\;}
\otimes L_i^{-1})$ and $H^{0}(X,L_j^{-1} \otimes L_i^{\;})$ cannot be nonzero.  It follows that $Q$ is acyclic. 

Let $Q$ be a quiver of sections associated to $\mathcal{L} = (L_0,
  \dotsc, L_r)$.

1.   By definition, $Q$ only depends on the line bundles $L_j^{\,}
        \otimes L_{i}^{-1}$ where $0 \leq i, j \leq r$. Consequently, for any line bundle $L'$ on $X$, we have $Q = Q'$ where $Q'$ is a quiver of sections associated to $\mathcal{L}' = (L_0 \otimes L',
        \dotsc, L_r \otimes L')$. To eliminate this redundancy, we will assume that $L_0 = \mathscr{O}_{X}$. By reordering the elements in $\mathcal{L}$ if necessary, we may also assume that $j < i$ implies $H^0(X,L_j^{\;} \otimes L_{i}^{-1}) = 0$.

2.   We will assume that $H^0(X,L_i) \neq 0$ for $0 \leq i \leq r$. If $Q$ is the complete quiver of sections for $\mathcal{L}$, then this implies that $Q$ is connected and rooted at $0 \in Q_0$.

Since each arrow $a \in Q_1$ corresponds to a $T_X$-invariant section $s \in H^0(X,L_j^{\;} \otimes L_{i}^{-1})$, we simply write ${\operatorname{div}}(a)
:= {\operatorname{div}}(s) \in \operatorname{CDiv}(X)$. More generally, for a path $p = a_1 \dotsb
a_\ell$ in $Q$, we set ${\operatorname{div}}(p) := {\operatorname{div}}(a_1) + \dotsb + {\operatorname{div}}(a_\ell)$. This labelling of paths induces relations on $Q$. Specifically, the *ideal of relations* is the two-sided ideal $R$ in the path algebra ${\ensuremath{\Bbbk}}Q$ generated by differences $p-p' \in {\ensuremath{\Bbbk}}Q$ such that $\operatorname{tl}(p) = \operatorname{tl}(p')$, $\operatorname{hd}(p) = \operatorname{hd}(p')$ and ${\operatorname{div}}(p) =
{\operatorname{div}}(p')$. Since the arrows in $Q$ correspond to indecomposable sections, $R$ is an admissible ideal.  The pair $(Q,R)$ is called a *bound quiver of sections*; the phrase “bound quiver” is a synonym for “quiver with relations”. 

\[exa:2pt\] If $L_1$ is a nontrivial line bundle on $X$, then the complete quiver of sections for $\mathcal{L} = (\mathscr{O}_{X}, L_1)$ has two vertices and $\dim_{{\ensuremath{\Bbbk}}} H^0(X,L_1)$ arrows.  The ideal of relations $R$ is the zero ideal. 

The general correspondence between bound quivers and finite-dimensional ${\ensuremath{\Bbbk}}$-algebras has the following useful incarnation for a quiver of sections. 

\[pro:algebra\] If $(Q,R)$ is the complete bound quiver of sections for $\mathcal{L}
  \!=\! ( L_0, \dotsc, L_r)$ then the quotient algebra ${\ensuremath{\Bbbk}}Q/R$ is isomorphic to $\operatorname{End}\bigl( \bigoplus_{i=0}^{r} L_i \bigr)$.

The map sending a path $p = a_1 \dotsb a_\ell$ in $Q$ to the product of the corresponding sections $s_1 \dotsb s_\ell \in
  H^0(X,L_{\operatorname{hd}(p)}^{\;} \otimes L_{\operatorname{tl}(p)}^{-1})=
  \operatorname{Hom}(L_{\operatorname{tl}(p)}, L_{\operatorname{hd}(p)})$ determines a homomorphism of ${\ensuremath{\Bbbk}}$-algebras $\eta \colon {\ensuremath{\Bbbk}}Q \rTo \operatorname{End}\bigl( \bigoplus_{i=0}^{r}
  L_i \bigr) = \bigoplus_{i,j=0}^{r} \operatorname{Hom}(L_i,L_j)$. The map is surjective because $Q$ is a complete quiver.  Moreover, $\eta$ sends paths $p, p'$ in $Q$ satisfying $\operatorname{tl}(p) = \operatorname{tl}(p')$ and $\operatorname{hd}(p)
  = \operatorname{hd}(p')$ to the same element in $\operatorname{Hom}(L_{\operatorname{tl}(p)},
  L_{\operatorname{hd}(p)})$ if and only if ${\operatorname{div}}(p) = {\operatorname{div}}(p')$. Thus, we have $\operatorname{Ker}(\eta) = R$.

Let $(Q,R)$ be a (not necessarily complete) bound quiver of sections associated to $\mathcal{L} = (L_0, \dotsc, L_r)$. If $\{ e_0,
  \dotsc, e_r \}$ is a complete set of primitive orthogonal idempotents in $\operatorname{End}\bigl( \bigoplus_{i=0}^{r} L_i \bigr)$ and $s_0,
  \dotsc, s_m$ are the indecomposable sections corresponding to the arrows in $Q$, then ${\ensuremath{\Bbbk}}Q / R$ is the subalgebra of $\operatorname{End}\bigl(
  \bigoplus_{i=0}^{r} L_i \bigr)$ generated by $\{ e_0, \dotsc, e_r,
  s_0, \dotsc, s_m \}$.

[@Bondal; @King1; @BergmanProudfoot] work with the opposite quiver.  In particular, a “Bondal quiver” is a complete quiver of sections in which arrows have the opposite orientation. 

A quiver of sections $Q$ comes equipped with a distinguished lattice.  The map sending $a \in Q_1$ to ${\operatorname{div}}(a) \in \operatorname{CDiv}(X)$ extends to give a ${\ensuremath{\mathbb{Z}}}$-linear map ${\operatorname{div}}\colon {\ensuremath{\mathbb{Z}}}^{Q_1} \rTo \operatorname{CDiv}(X)$ where ${\operatorname{div}}(v)
:= \sum_{a \in Q_1} v_{a} {\operatorname{div}}(a)$ for $v = \sum_{a \in Q_1} v_a
\chi_{a}$. The *section lattice* ${\ensuremath{\mathbb{Z}}}(Q)$ is the image of the map $\pi := (\operatorname{inc},{\operatorname{div}}) \colon {\ensuremath{\mathbb{Z}}}^{Q_{1}} \rTo \operatorname{Wt}(Q) \oplus
\operatorname{CDiv}(X)$; by definition, we have $\pi(\chi_a) = \bigl(
\chi_{\operatorname{hd}(a)} - \chi_{\operatorname{tl}(a)}, {\operatorname{div}}(a) \bigr)$. The projections onto the components are denoted by $\pi_{1} \colon {\ensuremath{\mathbb{Z}}}(Q) \rTo \operatorname{Wt}(Q)$ and $\pi_{2} \colon {\ensuremath{\mathbb{Z}}}(Q) \rTo \operatorname{CDiv}(X)$ respectively.  These maps fit in to the commutative diagram $$\label{eqn:rightsquare}
  \begin{diagram}[size=2em,midshaft,PS,labelstyle=\scriptstyle]
    {\ensuremath{\mathbb{Z}}}^{Q_1} & & & & \\
    & \rdTo\rdTo(4,2)^{\operatorname{inc}}\rdTo(2,4)_{{\operatorname{div}}} & & & \\
    & & {\ensuremath{\mathbb{Z}}}(Q) & \rTo_{\pi_1} & \operatorname{Wt}(Q) \\
    & & \dTo_{\pi_2} & & \dTo \\
    & & \operatorname{CDiv}(X) & \rTo & \operatorname{Pic}(X) \\
  \end{diagram}$$ where $\operatorname{pic}(\theta) := \bigotimes_{i \in Q_0} L_{i}^{\theta_{i}}$ for $\theta = \sum_{i \in Q_{0}} \theta_i \chi_{i}$.

\[exa:FF1\] Let $X = {\ensuremath{\mathbb{F}}}_{1} = {\ensuremath{\mathbb{P}}}\bigl( \mathscr{O}_{{\ensuremath{\mathbb{P}}}^1} \oplus
  \mathscr{O}_{{\ensuremath{\mathbb{P}}}^1}(1) \bigr)$ be the Hirzebruch surface determined by the fan in Figure \[fig:FF1\] (a). 

For $(k,\ell) \in {\ensuremath{\mathbb{Z}}}^2$, set $\mathscr{O}_{X}(k,\ell) :=
  \mathscr{O}_{X}(kD_1+\ell D_4) \in \operatorname{Pic}(X)$. The complete quiver of sections for $\mathcal{L} = \bigl( \mathscr{O}_{X},
  \mathscr{O}_{X}(1,0), \mathscr{O}_{X}(0,1) \bigr)$ appears in Figure \[fig:FF1\] (b).  The ideal of relations is $R = (0)$. The section lattice ${\ensuremath{\mathbb{Z}}}(Q)$ is generated by the columns of the matrix

$$\label{eq:FF1matrix}
    \left[ \text{\footnotesize $\begin{array}{rrrr}
        -1 & 0 & -1 & -1 \\
        1 & -1 & 1 & 0 \\
        0 & 1 & 0 & 1 \\
        1 & 0 & 0 & 0 \\
        0 & 1 & 0 & 0 \\
        0 & 0 & 1 & 0 \\
        0 & 0 & 0 & 1
      \end{array}$} \right] \, .$$

\[exa:FF2\] Let $X = {\ensuremath{\mathbb{F}}}_{2} = {\ensuremath{\mathbb{P}}}\bigl( \mathscr{O}_{{\ensuremath{\mathbb{P}}}^1} \oplus
  \mathscr{O}_{{\ensuremath{\mathbb{P}}}^1}(2) \bigr)$ be the Hirzebruch surface determined by the fan in Figure \[fig:FF2\] (a). 

For $(k,\ell) \in {\ensuremath{\mathbb{Z}}}^2$, we write $\mathscr{O}_{X}(k,\ell) :=
  \mathscr{O}_{X}(kD_1+\ell D_4) \in \operatorname{Pic}(X)$. The complete quiver of sections for $\mathcal{L} = \bigl( \mathscr{O}_X,
  \mathscr{O}_{X}(1,0), \mathscr{O}_{X}(0,1), \mathscr{O}_{X}(1,1)
  \bigr)$ appears in Figure \[fig:FF2\] (b).  If we order the arrows as in Figure \[fig:FF2\] (c), then the ideal of relations is $R =
  (a_2a_4-a_1a_5, a_4a_8-a_5a_7, a_2a_6-a_3a_8, a_1a_6-a_3a_7)$. The section lattice ${\ensuremath{\mathbb{Z}}}(Q)$ is generated by the columns of the following matrix

$$\label{eq:FF2matrix}
    \left[ \text{\footnotesize $\begin{array}{rrrrrrrr}
        -1 & -1 & -1 & 0 & 0 & 0 & 0 & 0 \\
        1 & 1 & 0 & -1 & -1 & -1 & 0 & 0 \\
        0 & 0 & 1 & 1 & 1 & 0 & -1 & -1 \\
        0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 \\ 
        1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 \\
        0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 \\
        0 & 1 & 0 & 0 & 1 & 0 & 0 & 1 \\
        0 & 0 & 1 & 0 & 0 & 1 & 0 & 0
      \end{array}$} \right] \, ;$$

the $i$-th column corresponds to $a_i$ for $1 \leq i \leq 8$.

Classically, one associates a projective space, called the linear series, to a nonzero subspace of global sections of a line bundle.  Generalizing this construction, we associate a toric variety $Y_Q$ to an appropriate quiver of sections $Q$. Any connected, rooted, acyclic quiver $Q$ defines a complete fan $\Sigma_Q$ in the ${\ensuremath{\mathbb{R}}}$-vector space $\operatorname{Hom}_{{\ensuremath{\mathbb{Z}}}}\bigl( \operatorname{Cir}(Q), {\ensuremath{\mathbb{R}}}\bigr)$. The rays $\Sigma_Q(1)$ correspond to arrows $a \in Q_1$ and are generated by the evaluation maps $\operatorname{ev}_{a} \colon \operatorname{Cir}(Q) \rTo {\ensuremath{\mathbb{R}}}$ defined by $\operatorname{ev}_a(f) = f_a$ for $f \in \operatorname{Cir}(Q)$. Thus, $\operatorname{ev}_{a}$ is the unique generator of $\rho_a \cap N_Y$ where $\rho_a \in \Sigma_Q(1)$ is the ray corresponding to $a \in Q_1$ and $N_Y := \operatorname{Hom}_{{\ensuremath{\mathbb{Z}}}}\bigl( \operatorname{Cir}(Q), {\ensuremath{\mathbb{Z}}}\bigr)$. The rays $\rho_{a_1}, \dotsc, \rho_{a_\ell} \in \Sigma_Q(1)$ span a cone in $\Sigma_Q$ if and only if there exists a spanning tree rooted at the source of $Q$ that does not contain $a_{1},\dots,
a_{\ell}$. Hence, maximal cones in $\Sigma_Q$ correspond to spanning trees rooted at the source, and have dimension $| Q_1|-| Q_0| + 1$. Since $\Sigma_Q$ can also be described as the triangulation associated to the region in the chamber complex corresponding to our acyclic orientation of the underlying graph of $Q$ (combine Theorem 3.1 in [@BGS] and Lemma 7.1 in [@GZ]), it follows that $\Sigma_Q$ is a fan.  Let $Y_Q$ be the toric variety determined by the fan $\Sigma_Q$.

The toric variety $Y_Q$, which is a toric quiver variety as defined in [@Hille1], has several other characterizations.  Following [@BPS], a toric variety $Y$ is unimodular if $M_Y$ is a unimodular sublattice of ${\ensuremath{\mathbb{Z}}}^{\Sigma_Y(1)}$; see .  This is equivalent to saying that $Y$ is smooth and any other variety obtained from $Y$ by toric flips and flops is also smooth. 

\[pro:smooth\] Let $Q$ be a connected, rooted, acyclic quiver.  If $Q_0 = \{ 0,
  \dotsc, r \}$ where $0$ is the unique source then the following varieties coincide:

1.   the toric variety $Y_Q$ defined by the fan $\Sigma_Q$;

2.   the geometric quotient of ${\ensuremath{\mathbb{A}}}^{Q_1} \setminus
        \mathbb{V}(B_Y)$ by the group $G = \operatorname{Hom}_{{\ensuremath{\mathbb{Z}}}}(\operatorname{Wt}(Q), {\ensuremath{\Bbbk}}^{*})$, where ${\ensuremath{\mathbb{A}}}^{Q_1} = \operatorname{Spec}(S_Y)$, $S_Y := {\ensuremath{\Bbbk}}[y_a : a \in Q_1]$ and $$B_Y := \Biggl( \prod\limits_{a \in Q'_1} y_a : \text{$Q'$ is a
          spanning tree of $Q$ rooted at $0$} \Biggl) = \bigcap\limits_{i =
          1}^{r} \bigl( y_a : \operatorname{hd}(a) = i \bigr) \, ;$$

3.   the fine moduli space $\mathcal{M}_\theta(Q)$ of $\theta$-stable representations for any rational weight $\theta
        \in \operatorname{Wt}(Q) \otimes_{{\ensuremath{\mathbb{Z}}}}{\ensuremath{\mathbb{Q}}}$ lying in the open GIT-chamber containing $\vartheta:=\sum_{i\in Q_0} (\chi_i-\chi_0)$.

Moreover, this variety is unimodular and projective. 

Let $W$ be a $\vartheta$-semistable representation of $Q$. If $W'
  \subset W$ is a proper nonzero subrepresentation, then we have $\vartheta(W') = \sum_{i \in \operatorname{supp}(W')} \vartheta_i \geq 0$. Since $\vartheta_i = 1$ for $i \neq 0$ and $\vartheta_0 = -r$, it follows that $W_0' = 0$ and $\vartheta(W') > 0$. Therefore, $\vartheta$ is generic, the open chamber containing $\vartheta$ is well-defined, and results in §4 of [@King1] show that $\mathcal{M}_\vartheta(Q)$ is a smooth projective variety.  Since the map $\operatorname{inc}\colon {\ensuremath{\mathbb{Z}}}^{Q_1}\rTo \operatorname{Wt}(Q)$ is totally unimodular (e.g. Proposition 5.3 in [@Biggs] or Example 2 in §19.3 of [@Sch]), it follows that $\operatorname{Cir}(Q)$ is a unimodular sublattice of ${\ensuremath{\mathbb{Z}}}^{Q_1}$ and the toric variety defined by $\Sigma_Q$ is unimodular.  Theorem 2.1 in [@Cox] establishes the equivalence between (a) and (b), and the discussion preceding Theorem 1.7 in [@Hille1] establishes the equivalence between (a) and (c). 

When $Q$ is a connected, rooted, acyclic quiver of sections on $X$, the toric variety $Y_Q$ is called the *multilinear series* of $Q$. When the quiver of sections is unambiguous, we simply write $Y$ for the multilinear series.  If $Q$ is the complete quiver of sections for a list a of line bundles $\mathcal{L}$, then we write $|
\mathcal{L}| := Y$ for the *complete multilinear series*. 

\[rem:Y\] Proposition \[pro:smooth\] implies that the diagram becomes $$\begin{diagram}[size=2em,l>=2em,midshaft,labelstyle=\scriptstyle]
      0 & \rTo & M_Y & \rTo & \operatorname{CDiv}(Y) & \rTo & \operatorname{Pic}(Y) & \rTo & 0 \,
      \, \\
      & & \dTo & & \dTo & & \dTo & & \\
      0 & \rTo & \operatorname{Cir}(Q) & \rTo & {\ensuremath{\mathbb{Z}}}^{Q_1} & \rTo^{\operatorname{inc}} & \operatorname{Wt}(Q) &
      \rTo & 0 \, ,
    \end{diagram}$$ where the vertical maps are isomorphisms.  The open chamber containing $\vartheta$ is the ample cone $\operatorname{Amp}_{\ensuremath{\mathbb{Q}}}(Y)$ of ${\ensuremath{\mathbb{Q}}}$-divisor classes on $Y$ and the closure of this chamber is $$\operatorname{Nef}_{\ensuremath{\mathbb{Q}}}(Y) = \bigcap\limits_{Q' \subseteq Q} \Biggl\{ \sum\limits_{a
    \in Q_1'} \lambda_a [D_a] : \lambda_a \in {\ensuremath{\mathbb{Q}}}_{\geq 0} \Biggr\} \,
  ,$$ where the intersection runs over all spanning trees $Q'$ of $Q$ rooted at $0$ and $D_a$ is the irreducible $T_Y$-invariant Weil divisor associated to $a \in Q_1$. Since $Y$ is smooth, the ample line bundle $\mathscr{O}_{Y}(\vartheta)$ determined by $\vartheta
  \in \operatorname{Wt}(Q)$ is very ample. 

\[exa:2ptY\] Let $Q$ be a quiver with $Q_0 = \{ 0, 1 \}$ and $Q_1 = \{ a_0,
  \dotsc, a_m \}$ such that $\operatorname{tl}(a_k) = 0$ and $\operatorname{hd}(a_k) = 1$ for all $0 \leq k \leq m$. Since every arrow forms a spanning tree rooted at $0$, the irrelevant ideal of $Y$ is $B_Y = ( y_{a_0},
  \dotsc, y_{a_{m}})$. Hence, $Y$ is the geometric quotient of ${\ensuremath{\mathbb{A}}}^{Q_1} \setminus \{ 0 \}$ by $G := \operatorname{Hom}_{{\ensuremath{\mathbb{Z}}}}\bigl( \operatorname{Wt}(Q),
  {\ensuremath{\Bbbk}}^{*})$. Choosing $\chi_1 - \chi_0$ as a basis for $\operatorname{Wt}(Q)$, we see that $G \cong {\ensuremath{\Bbbk}}^*$ and the $G$-action is induced by the matrix is $\left[ \begin{smallmatrix} 1 & \dotsb & 1 \end{smallmatrix}
  \right]$. Therefore, we have $Y \cong {\ensuremath{\mathbb{P}}}^{m}$. In particular, if $Q$ is the complete quiver of sections for $\mathcal{L} =
  (\mathscr{O}_{X}, L_1)$ described in Example \[exa:2pt\], then the complete multilinear series $|\mathcal{L}|$ is canonically isomorphic to the linear series $|L_1|$.

\[exa:FF1Y\] Let $X = {\ensuremath{\mathbb{F}}}_1$ and $\mathcal{L} = \bigl( \mathscr{O}_X,
  \mathscr{O}_{X}(1,0), \mathscr{O}_{X}(0,1) \bigr)$ as in Example \[exa:FF1\].  If we identify $\operatorname{Cir}(Q)$ with ${\ensuremath{\mathbb{Z}}}^2$ by choosing the circuits $(a_1^{\,}a_3^{-1}, a_3^{\,}a_2^{\,}a_4^{-1})$ as an ordered basis, then the unique generator of $\rho_{k} \cap
  N_Y$, where $\rho_k \in \Sigma_Q(1)$ corresponds to $a_k \in Q_1$, is the $k$-th column of the matrix $\bigl[
    \text{\footnotesize $\begin{array}{rrrr}
      1 & 0 & -1 & 0 \\ 0 & 1 & 1 & -1
    \end{array}$} \bigr]$. Figure \[fig:FF1\] (c) gives $S_Y = {\ensuremath{\Bbbk}}[y_1, \dotsc, y_4]$ and $B_Y = (y_1,y_3) \cap (y_2, y_4)$. Hence, the quotient construction of $Y$ from Proposition \[pro:smooth\] (b) coincides with the quotient construction of $X$ encoded by the pair $(S_X,B_X)$; see §\[sub:toric\].  Therefore, the multilinear series $Y$ equals $X$.

\[exa:FF2Y\] Let $X = {\ensuremath{\mathbb{F}}}_2$ and $\mathcal{L} = \bigl( \mathscr{O}_X,
  \mathscr{O}_{X}(1,0), \mathscr{O}_{X}(0,1), \mathscr{O}_{X}(1,1)
  \bigr)$ as in Example \[exa:FF2\].  If we identify $\operatorname{Cir}(Q)$ with ${\ensuremath{\mathbb{Z}}}^5$ by choosing the circuits $$(a_1^{\,}a_2^{-1}, a_1^{\,}a_4^{\,}a_3^{-1},
  a_1^{\,}a_5^{\,}a_3^{-1}, a_1^{\,}a_6^{\,}a_7^{-1}a_3^{-1},
  a_1^{\,}a_6^{\,}a_8^{-1}a_3^{-1})$$ as an ordered basis, then the unique generator of $\rho_{k} \cap
  N_Y$, where $\rho_k \in \Sigma_Q(1)$ corresponds to $a_k \in Q_1$, is the $k$-th column of the matrix

$$\left[ 
    \text{\footnotesize $\begin{array}{rrrrrrrr}
      1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 \\
      1 & 0 & -1 & 1 & 0 & 0 & 0 & 0 \\
      1 & 0 & -1 & 0 & 1 & 0 & 0 & 0 \\
      1 & 0 & -1 & 0 & 0 & 1 & -1 & 0 \\
      1 & 0 & -1 & 0 & 0 & 1 & 0 & -1 
    \end{array}$} \right] \, .$$ Figure \[fig:FF2\] (c) implies that $S_Y = {\ensuremath{\Bbbk}}[y_1, \dotsc, y_8]$ and $B_Y = (y_1,y_2) \cap (y_3, y_4, y_5) \cap (y_6, y_7, y_8)$. Hence, the multilinear series $Y$ is a smooth $5$-dimensional toric variety with $8$ irreducible $T_Y$-invariant Weil divisors and $18$ $T_Y$-fixed points.  The ample cone of $Y$ is $$\operatorname{Amp}_{\ensuremath{\mathbb{Q}}}(Y) = \left\{ \theta = (\theta_0, \theta_1,
    \theta_2, \theta_3) \in \operatorname{Wt}(Q) \otimes_{{\ensuremath{\mathbb{Z}}}} {\ensuremath{\mathbb{Q}}}: \text{$\theta_1
      > 0$, $\theta_2 > 0$, $\theta_3 > 0$} \right\} \, .$$ Since $(-3,-1,1,3) \not\in \operatorname{Amp}_{\ensuremath{\mathbb{Q}}}(Y)$, the dualizing line bundle on $Y$ is not ample. 

Multilinear Series {#sec:multilinear}
==================

In this section, we study morphisms from the toric variety $X$ to the multilinear series $Y$ induced by a quiver of sections.  To begin, we give necessary and sufficient conditions for a quiver of sections $Q$ on $X$ to define a morphism from $X$ to the multilinear series $Y$. The divisors labelling the arrows in $Q$ define a ring homomorphism $\Phi_Q \colon S_{Y} \rTo S_X$ between the total coordinate rings of $X$ and $Y$ given by $\Phi_Q(y_a) = x^{{\operatorname{div}}(a)}$. The *base ideal* of $Q$ is the ideal $B_Q$ in $S_X$ generated by the image $\Phi_Q(B_Y)$.

\[pro:free\] Let $Q$ be a connected, rooted, acyclic quiver of sections on $X$. If $Y$ is the multilinear series of $Q$, then the following are equivalent:

1.   the map $\Phi_Q$ determines a morphism $\varphi_Q
        \colon X \rTo Y$;

2.   the irrelevant ideal $B_X$ is contained in the radical $\operatorname{rad}(B_Q)$;

3.   for all $\sigma \in \Sigma_X$, there exists a spanning tree $Q' \subseteq Q$ rooted at the unique source in $Q$ such that $\operatorname{supp}({\operatorname{div}}(a)) \subseteq \widehat{\sigma}$ for all $a \in Q'_1$.

The toric variety $X$ is a categorical quotient of ${\ensuremath{\mathbb{A}}}^{\Sigma_{X}(1)} \setminus {\ensuremath{\mathbb{V}}}(B_X)$ under the action of the group $\operatorname{Hom}_{{\ensuremath{\mathbb{Z}}}}(\operatorname{Cl}(X), {\ensuremath{\Bbbk}}^*)$; see §\[sub:toric\].  Similarly, Proposition \[pro:smooth\] shows that $Y$ is the geometric quotient of ${\ensuremath{\mathbb{A}}}^{Q_1} \setminus {\ensuremath{\mathbb{V}}}(B_Y)$ under the action of the group $G = \operatorname{Hom}_{{\ensuremath{\mathbb{Z}}}}(\operatorname{Pic}(Y), {\ensuremath{\Bbbk}}^*)$. The ring map $\Phi_Q \colon S_Y \rTo S_X$ defines a morphism from ${\ensuremath{\mathbb{A}}}^{\Sigma_{X}(1)}$ to ${\ensuremath{\mathbb{A}}}^{Q_1}$. This morphism is equivariant with respect to the actions of the groups $\operatorname{Hom}_{{\ensuremath{\mathbb{Z}}}}(\operatorname{Cl}(X),
  {\ensuremath{\Bbbk}}^*)$ and $G$ on ${\ensuremath{\mathbb{A}}}^{\Sigma_{X}(1)}$ and ${\ensuremath{\mathbb{A}}}^{Q_1}$ respectively because the lattice maps $$\begin{diagram}[size=2em,l>=2em,midshaft,labelstyle=\scriptstyle]
    {\ensuremath{\mathbb{Z}}}^{Q_1} & \rTo^{\operatorname{inc}} & \operatorname{Wt}(Q) \\
    \dTo^{{\operatorname{div}}} & &  \dTo_{[\operatorname{pic}]} \\
    {\ensuremath{\mathbb{Z}}}^{\Sigma_X(1)} & \rTo & \operatorname{Cl}(X)  \,     
  \end{diagram}$$ commute.  Thus, $\Phi_Q$ induces the morphism $\varphi_Q \colon X
  \rTo Y$ if and only if the preimage of the irrelevant set ${\ensuremath{\mathbb{V}}}(B_Y)$ is contained in the irrelevant set ${\ensuremath{\mathbb{V}}}(B_X)$; see Theorem 3.2 in [@Cox2].  Since the preimage of ${\ensuremath{\mathbb{V}}}(B_Y)$ is cut out by the base ideal $B_Q$, this is equivalent to $B_X$ being contained in $\operatorname{rad}(B_Q)$. In other words, conditions (a) and (b) are equivalent.  The definition of $B_X$ and the explicit description of $B_Y$ from Proposition \[pro:smooth\] (b) gives the equivalence between (b) and (c). 

A quiver of sections $Q$ is *basepoint-free* if it is connected, rooted, acyclic, and satisfies any of the equivalent conditions in Proposition \[pro:free\].  If the complete quiver of sections for $\mathcal{L}$ is basepoint-free, then $\varphi_{|\mathcal{L}|} \colon
X \rTo |\mathcal{L}|$ denotes the associated morphism to the complete multilinear series. 

\[cor:basepointfree\] If $Q$ is a basepoint-free quiver of sections then each line bundle $L_i$ on $X$ is basepoint-free.  Conversely, if each $L_i$ is basepoint-free and $Q$ is a complete quiver of sections for $\mathcal{L} = (\mathscr{O}_X,L_1, \dotsc, L_r)$ then $Q$ is basepoint-free. 

Since $Q$ is basepoint-free, it satisfies condition (c) from Proposition \[pro:free\].  Hence, for $i \in Q_0$ and $\sigma \in
  \Sigma_X(d)$, there exists a path $p = a_1 \dotsc a_\ell$ in $Q$ such that $\operatorname{tl}(p) = 0$, $\operatorname{hd}(p) = i$, and $\operatorname{supp}\bigl({\operatorname{div}}(a_k)\bigr) \subseteq \widehat{\sigma}$ for all $1
  \leq k \leq \ell$. In other words, $L_i$ admits a $T_X$-invariant section that does not vanish at the $T_X$-fixed point indexed by $\sigma$ for all $\sigma \in
  \Sigma_X(d)$. Therefore, for each $i \in Q_0$, $L_i$ is basepoint-free.  When $Q$ is complete, we can reverse the argument for the first part. 

Given a basepoint-free quiver of sections $Q$, the image of $\varphi_Q$ can be described explicitly.  Let ${\ensuremath{\mathbb{N}}}(Q)$ be the image of ${\ensuremath{\mathbb{N}}}^{Q_1}$ under the map $\pi \colon {\ensuremath{\mathbb{Z}}}^{Q_1} \rTo \operatorname{Wt}(Q) \oplus
\operatorname{CDiv}(X)$. Observe that ${\ensuremath{\mathbb{N}}}(Q)$ is a subsemigroup of the section lattice ${\ensuremath{\mathbb{Z}}}(Q)$. Since $S_Y$ is the semigroup algebra of ${\ensuremath{\mathbb{N}}}^{Q_1}$, the map $\pi$ induces a surjective ${\ensuremath{\Bbbk}}$-algebra homomorphism from $S_Y$ to ${\ensuremath{\Bbbk}}[{\ensuremath{\mathbb{N}}}(Q)]$ where ${\ensuremath{\Bbbk}}[{\ensuremath{\mathbb{N}}}(Q)]$ is the semigroup algebra of ${\ensuremath{\mathbb{N}}}(Q)$. The kernel of this induced map is the toric ideal $I_Q := \bigl( y^u - y^v \in S_Y : u - v \in \operatorname{Ker}(\pi)
\bigr)$. The $G$-invariant affine toric variety ${\ensuremath{\mathbb{V}}}(I_Q) \subseteq
{\ensuremath{\mathbb{A}}}^{Q_1}$ cut out by the ideal $I_Q$ need not be normal.  The ideal $I_Q$ is analogous to the toric ideal defined by the augmented vertex-edge incidence matrix of the McKay quiver in [@CMT2].  With this notation, we obtain the following. 

\[pro:ideal\] Let $Q$ be a basepoint-free quiver of sections on $X$. If $\varphi_Q \colon X \rTo Y$ is the induced morphism, then the image of $\varphi_Q$ is:

1.   the subscheme of $Y$ corresponding to the $\operatorname{Wt}(Q)$-graded $B_Y$-saturated ideal $I_Q$;

2.   the geometric quotient of ${\ensuremath{\mathbb{V}}}(I_Q) \setminus {\ensuremath{\mathbb{V}}}(B_Y)$ by $G
        = \operatorname{Hom}_{\ensuremath{\mathbb{Z}}}\bigl( \operatorname{Wt}(Q), {\ensuremath{\Bbbk}}^* \bigr)$;

3.   the GIT-quotient ${\ensuremath{\mathbb{V}}}(I_Q) {\ensuremath{/\!\!/_{\!\theta}\,}}G$ for any $\theta \in
        \operatorname{Amp}_{\ensuremath{\mathbb{Q}}}(Y)$.

Proposition \[pro:smooth\] (b) implies that the closed subsets of $Y$ are in bijection with the $G$-invariant closed subsets of ${\ensuremath{\mathbb{A}}}^{Q_1} \setminus {\ensuremath{\mathbb{V}}}(B_Y)$, and hence with the $B_Y$-saturated ideals of $S_Y$. The image of the map from $\mathbb{A}^{\Sigma_X(1)}$ to $\mathbb{A}^{Q_1}$ induced by the homomorphism of semigroups ${\operatorname{div}}\colon \mathbb{N}^{Q_1} \rTo
  \operatorname{CDiv}(X)$ is cut out by the toric ideal $\operatorname{Ker}(\Phi_Q)$. Since the action of $G = \operatorname{Hom}_{{\ensuremath{\mathbb{Z}}}}\bigl(\operatorname{Wt}(Q),{\ensuremath{\Bbbk}}^*\bigr)$ on ${\ensuremath{\mathbb{A}}}^{Q_1}$ is induced by the map $\operatorname{inc}\colon {\ensuremath{\mathbb{Z}}}^{Q_1} \rTo \operatorname{Wt}(Q)$, the toric ideal $I_Q$ associated to the map $\pi = (\operatorname{inc},{\operatorname{div}}) \colon
  {\ensuremath{\mathbb{N}}}^{Q_1} \rTo \operatorname{Wt}(Q) \oplus \operatorname{CDiv}(X)$ cuts out the $\operatorname{Wt}(Q)$-homogeneous part of $\operatorname{Ker}(\Phi_Q)$. As $I_Q$ is prime and hence $B_Y$-saturated, (a) and (b) follow.  The equivalence of (b) and (c) follows directly from the equivalence between (b) and (c) in Proposition \[pro:smooth\]. 

\[rem:chamber\] Proposition \[pro:ideal\] (c) holds for rational weights in a cone that may strictly contain $\operatorname{Amp}_{\ensuremath{\mathbb{Q}}}(Y)$ because the GIT-chamber decomposition for the $G$-action on ${\ensuremath{\mathbb{V}}}(I_Q)$ is a coarsening of that for the $G$-action on ${\ensuremath{\mathbb{A}}}^{Q_1}$.

The complete quiver of sections for $\mathcal{L} = (\mathscr{O}_X,
  L_1, \dotsc, L_r)$ is basepoint-free by Corollary \[cor:basepointfree\], so $\varphi_{|\mathcal{L}|}$ is a morphism.  The explicit description of the image as a geometric quotient is presented in Proposition \[pro:ideal\] (b). 

\[exa:2ptimage\] Corollary \[cor:basepointfree\] show that the complete quiver of sections for $\mathcal{L} = (\mathscr{O}_X, L_1)$ is basepoint-free if and only if $L_1$ is basepoint-free.  Since the semigroup ${\ensuremath{\mathbb{N}}}(Q)$ is generated by $\bigl( (-1, 1), {\operatorname{div}}(s) \bigr) \in {\ensuremath{\mathbb{Z}}}^2
  \oplus \operatorname{CDiv}(X)$ for $T_X$-invariant sections $s \in H^0(X, L_1)$, it is isomorphic to the semigroup generated by the effective divisors ${\operatorname{div}}(s)$ in $\operatorname{CDiv}(X)$. Hence $I_Q$ is the ideal for the image of $X$ in $|L_1| := {\ensuremath{\mathbb{P}}}\bigl( H^0(X,L_1) \bigr)$. It follows that ${\ensuremath{\mathbb{V}}}(I_Q)$ is the affine cone over $\varphi_{|L_1|}(X)$, and $\varphi_{|\mathcal{L}|} = \varphi_{|L_1|}$.

\[exa:FF1image\] Let $X = {\ensuremath{\mathbb{F}}}_1$ and $\mathcal{L} = \bigl( \mathscr{O}_X,
  \mathscr{O}_{X}(1,0), \mathscr{O}_{X}(0,1) \bigr)$ as in Example \[exa:FF1\].  Example \[exa:FF1Y\] shows that the multilinear series is $Y = |\mathcal{L}| = X$. By examining Figure \[fig:FF1\] (b), we see that the complete quiver of sections for $\mathcal{L}$ satisfies condition (c) in Proposition \[pro:free\].  By definition, ${\ensuremath{\mathbb{N}}}(Q)$ is generated by the columns of the matrix in so the toric ideal is $I_Q = (0) \subset S_Y$. Thus, $\varphi_{|\mathcal{L}|}$ is an isomorphism. 

\[exa:FF2image\] Let $X = {\ensuremath{\mathbb{F}}}_2$ and $\mathcal{L} = \bigl( \mathscr{O}_X,
  \mathscr{O}_{X}(1,0), \mathscr{O}_{X}(0,1), \mathscr{O}_{X}(1,1)
  \bigr)$ as in Example \[exa:FF2\].  The multilinear series $Y =
  |\mathcal{L}|$ is described in Example \[exa:FF2Y\].  By examining Figure \[fig:FF2\] (b), we see that the complete quiver of sections for $\mathcal{L}$ satisfies condition (c) in Proposition \[pro:free\].  By definition, ${\ensuremath{\mathbb{N}}}(Q)$ is generated by the columns of the matrix in , so $I_Q =
  (y_2y_4-y_1y_5, y_1y_6-y_3y_7, y_2y_6-y_3y_8, y_2y_7-y_1y_8,
  y_5y_7-y_4y_8) \subset S_Y$. In this case, we have $\varphi_{|\mathcal{L}|}(X) = {\ensuremath{\mathbb{V}}}(I_Q) {\ensuremath{/\!\!/_{\!\theta}\,}}G$ for all rational weights $\theta$ in the GIT-chamber $$\Theta := \bigl\{ (\theta_0,\theta_1,\theta_2,\theta_3) \in \operatorname{Wt}(Q)
  \otimes_{\ensuremath{\mathbb{Z}}}{\ensuremath{\mathbb{Q}}}: 
  \theta_3 > 0,\; \theta_1 + \theta_3 > 0,\; \theta_2 + \theta_3>0,\;
  \theta_1+\theta_2+\theta_3>0 \bigr\} \, ;$$ see Remark \[rem:chamber\]. 

With additional hypotheses, we can enlarge the commutative diagram . 

\[cor:funddiagram\] If $Q$ is basepoint-free and $\dim \varphi_Q(X) = \dim X$, then $$\begin{diagram}[size=2em,l>=2em,midshaft,labelstyle=\scriptstyle]
      0 & \rTo & M_X & \rTo & {\ensuremath{\mathbb{Z}}}(Q) & \rTo^{\pi_1} & \operatorname{Wt}(Q) & \rTo &
      0 \\
      & & \dEqual & & \dTo_{\pi_2} & & \dTo_{\operatorname{pic}} & & \\
      0 & \rTo & M_X & \rTo & \operatorname{CDiv}(X) & \rTo & \operatorname{Pic}(X) & \rTo & 0 \\ 
    \end{diagram}$$ is a commutative diagram with exact rows.  In particular, the projection $\pi_2$ induces an isomorphism between $\operatorname{Ker}(\pi_1)$ and $M_X$.

Combining with the top row of , it is enough to prove that $\pi_2$ yields an isomorphism between $\operatorname{Ker}(\pi_1)$ and $M_X$. The morphism $\varphi_Q \colon X \rTo \varphi_Q(X)$ corresponds to the map of semigroup algebras $S_Y/I_Q = {\ensuremath{\Bbbk}}[\mathbb{N}(Q)] \rTo
  {\ensuremath{\Bbbk}}[\mathbb{N}^{\Sigma_X(1)}] = S_X$ induced by $\pi_2$. Since $\dim \varphi_Q(X) = \dim X$, it identifies the dense tori in $X$ and $\varphi_Q(X)$. Therefore, $\pi_2$ identifies the character lattices $M_X$ and $\operatorname{Ker}(\pi_1)$.

Next, we give a criterion for $\varphi_Q \colon X \rTo Y$ to be a closed embedding.  For $\sigma \in \Sigma_X(d)$, let $y^{\widehat{\sigma}} := \prod_{\operatorname{supp}({\operatorname{div}}(a)) \subseteq
  \widehat{\sigma}} y_a$ be the associated monomial in $S_Y$. The localization of an $S_Y$-module $F$ at the element $y^{-
  \widehat{\sigma}}$ is denoted by $F[y^{- \widehat{\sigma}}]$. The weight $\vartheta:=\sum_{i\in Q_0} (\chi_i-\chi_0)$ appearing below is defined in Proposition \[pro:smooth\] (c). 

\[pro:ample\] Let $Q$ be a basepoint-free quiver of sections.  The map $\varphi_Q
  \colon X \rTo Y$ is a closed embedding if and only if the line bundle $L := L_0^{\vartheta_0} \otimes \dotsb \otimes L_r^{\vartheta_r} =
  \bigotimes_{i \in Q_0} L_i$ is ample and $\bigl( (S_Y/I_Q)[y^{-
    \widehat{\sigma}}] \bigr)_{[0]} \cong \bigl( S_X[x^{-
    \widehat{\sigma}}] \bigr)_{[0]}$ for all $\sigma \in \Sigma_X(d)$.

The very ample line bundle $\mathscr{O}_{Y}(\vartheta)$ from Remark \[rem:Y\] provides a closed embedding $Y \rTo {\ensuremath{\mathbb{P}}}^{m} := {\ensuremath{\mathbb{P}}}\bigl( H^0(Y, \mathscr{O}_{Y}(\vartheta)) \bigr)$. Hence, $\varphi_Q \colon X \rTo Y$ is a closed embedding if and only if the composition $X \rTo {\ensuremath{\mathbb{P}}}^m$ is a closed embedding.  The map $X \rTo
  {\ensuremath{\mathbb{P}}}^m$ is determined by $\varphi_Q^{*}\bigl(
  \mathscr{O}_{Y}(\vartheta) \bigr) = \operatorname{pic}(\vartheta)= L$ and the subspace $\Phi_Q\bigl( (S_Y)_\vartheta \bigr) \subseteq (S_X)_{[L]}
  \cong H^0( X, L)$ of global sections.  The morphism $\varphi_Q
  \colon X \rTo Y$ corresponds to the map of semigroup algebras $S_Y/I_Q = {\ensuremath{\Bbbk}}[\mathbb{N}(Q)] \rTo {\ensuremath{\Bbbk}}[\mathbb{N}^{\Sigma_X(1)}] =
  S_X$ induced by $\pi_2 \colon {\ensuremath{\mathbb{Z}}}(Q) \rTo \operatorname{CDiv}(X)$ which implies that $(S_Y/I_Q)_\vartheta \cong \Phi_Q((S_Y)_\vartheta)$. Moreover, the set $\mathcal{V} := {\ensuremath{\mathbb{N}}}(Q)\cap \pi_1^{-1}(\vartheta)$ can be identified with both the monomial basis of $(S_Y/I_Q)_\vartheta$ and a subset of lattice points in the polytope associated to $L$. Since $Q$ is basepoint-free, Proposition \[pro:free\] (c) implies that, for each cone $\sigma \in \Sigma_X(d)$, there exists a monomial $y^{u_{\sigma}} \in (S_Y)_\vartheta$ such that $x^{v_\sigma} :=
  \Phi_Q(y^{u_\sigma}) \in (S_X)_{[L]}$ satisfies $\operatorname{supp}(x^{v_\sigma})
  \subseteq \widehat\sigma$. By Theorem 2.7 in [@Oda], the monomial $x^{v_\sigma} \in S_X$ corresponds to the vertex $v_\sigma
  \in \mathcal{V}$ of the polytope associated to $L$. From the proof of the Theorem 2.13 in [@Oda], we deduce that $X \rTo {\ensuremath{\mathbb{P}}}^m$ is a closed embedding if and only if $v_\sigma \neq v_\tau$ holds for each pair $\sigma \neq \tau \in \Sigma_X(d)$, and the semigroup $M_X
  \cap \sigma^{\vee}$ is generated by $\mathcal{V} - v_\sigma$ for all $\sigma \in \Sigma_X(d)$. Corollary 2.14 in [@Oda] proves that the first condition is equivalent to $L$ being ample.  Lemma 2.2 in [@Cox] shows that ${\ensuremath{\Bbbk}}[M_X \cap \sigma^{\vee}] \cong
  (S_X[x^{-\widehat\sigma}])_{[0]}$. Working in the semigroup algebra ${\ensuremath{\Bbbk}}[{\ensuremath{\mathbb{N}}}(Q)] = S_Y/I_Q$ rather than ${\ensuremath{\Bbbk}}[{\ensuremath{\mathbb{N}}}^{\Sigma_X(1)}] = S_X$, essentially the same argument establishes that $\bigl(
  (S_Y/I_Q)[y^{- \widehat\sigma}] \bigr)_{[0]}$ is isomorphic to the semigroup algebra of $\mathcal{V} - v_{\sigma}$. Therefore, the second condition is equivalent to $\bigl( (S_Y/I_Q)[y^{-
    \widehat{\sigma}}] \bigr)_{[0]} \cong \bigl( S_X[x^{-
    \widehat{\sigma}}] \bigr)_{[0]}$ for all $\sigma \in \Sigma_X(d)$.

A quiver of sections $Q$ is *very ample* if it is complete, basepoint-free, and $\varphi_Q \colon X \rTo Y$ is a closed embedding.  For convenience, we record an instance of Proposition \[pro:ample\]. 

\[cor:veryample\] Let $\mathcal{L} = (\mathscr{O}_X, L_1, \dotsc, L_r)$ a list of basepoint-free line bundles and set $L := \bigotimes_{i \in Q_0}
  L_i$. Assume that the map $H^0(X,L_1) \otimes_{\ensuremath{\Bbbk}}\dotsb
  \otimes_{\ensuremath{\Bbbk}}H^0(X,L_r) \rTo H^0(X,L)$ is surjective.  The morphism $\varphi_{|\mathcal{L}|} \colon X \rTo Y$ is a closed embedding if and only if $L$ is very ample. 

Since each $L_i$ is basepoint-free, Corollary \[cor:basepointfree\] implies that the complete quiver of sections $Q$ for $\mathcal{L}$ is basepoint-free, so $(S_Y/I_Q)_{\chi_i - \chi_0} \cong (S_X)_{[L_i]} \cong H^0(X,L_i)$. Since $L_0 = \mathscr{O}_X$ and $\vartheta = \sum_{i=1}^{r} (\chi_i
  - \chi_0)$, we may identify the image of the map $$H^0(X,L_0^{\vartheta_0}) \otimes_{\ensuremath{\Bbbk}}\dotsb \otimes_{\ensuremath{\Bbbk}}H^0(X,L_r^{\vartheta_r}) = H^0(X,L_1) \otimes_{\ensuremath{\Bbbk}}\dotsb \otimes_{\ensuremath{\Bbbk}}H^0(X,L_r) \rTo H^0(X,L)$$ with the vector space $(S_Y/I_Q)_\vartheta$. From the proof of Proposition \[pro:ample\], we know that $\varphi_Q \colon X \rTo Y$ is a closed embedding if and only if the map to projective space, determined by the line bundle $L$ and the subspace $(S_Y/I_Q)_\vartheta \subseteq H^0(X,L)$, is a closed embedding.  The hypothesis that $H^0(X,L_1) \otimes_{\ensuremath{\Bbbk}}\dotsb \otimes_{\ensuremath{\Bbbk}}H^0(X,L_r) \rTo H^0(X,L)$ is surjective implies that $(S_Y/I_Q)_\vartheta \cong H^0(X,L)$. Lastly, we observe that the complete linear series $|L|$ determines a closed embedding if and only $L$ is very ample. 

Corollary \[cor:veryample\] shows that the complete quiver of sections for the list $\mathcal{L} = (\mathscr{O}_X, L_1)$ is very ample if and only the line bundle $L_1$ is very ample. 

\[exa:FF1ample\] Let $X = {\ensuremath{\mathbb{F}}}_2$ and $\mathcal{L} = \bigl( \mathscr{O}_X,
  \mathscr{O}_{X}(1,0), \mathscr{O}_{X}(0,1) \bigr)$ as in Example \[exa:FF1\].  Since $L_1 \otimes L_2 = \mathscr{O}_X(1,1)$ is very ample and the map $$H^0(X,L_1) \otimes_{\ensuremath{\Bbbk}}H^0(X,L_2) \rTo H^0 \bigl(X,
  \mathscr{O}_X(2,2) \bigr)$$ is surjective, the complete quiver of sections for $\mathcal{L}$ is very ample by Corollary \[cor:veryample\]. 

\[exa:FF2ample\] Let $X = {\ensuremath{\mathbb{F}}}_2$ and $\mathcal{L} = \bigl( \mathscr{O}_X,
  \mathscr{O}_{X}(1,0), \mathscr{O}_{X}(0,1), \mathscr{O}_{X}(1,1)
  \bigr)$ as in Example \[exa:FF2\].  Since the line bundle $L_1
  \otimes L_2 \otimes L_3 = \mathscr{O}_X(2,2)$ is very ample on $X$ and the map $H^0(X,L_1) \otimes_{\ensuremath{\Bbbk}}H^0(X,L_2) \otimes_{\ensuremath{\Bbbk}}H^0(X,L_3) \rTo H^0 \bigl(X, \mathscr{O}_X(2,2) \bigr)$ is surjective, Corollary \[cor:veryample\] implies that the complete quiver of sections for $\mathcal{L}$ is very ample. 

To see that every list of basepoint-free line bundles belongs to some very ample quiver of sections, we prove:

\[pro:veryample\] Let $L_1,\dotsc,L_{r-1}$ be basepoint-free line bundles on $X$. If the subsemigroup of $\operatorname{Pic}(X)$ generate by $L_1, \dotsc, L_{r-1}$ contains an ample line bundle, then there exists a line bundle $L_r$ such that the complete quiver of sections for $\mathcal{L} =
 (\mathscr{O}_X, L_1, \dotsc, L_r)$ is very ample. 

By choosing $b_1,\dotsc, b_{r-1} \in {\ensuremath{\mathbb{N}}}$ sufficiently large, we may assume that the line bundle $L_r := L_1^{b_1} \otimes \dotsb \otimes
  L_{r-1}^{b_{r-1}}$ is very ample and $\mathscr{O}_X$-regular with respect to $L_1, \dotsc, L_{r-1}$; for the multigraded definition of regularity see [@MaclaganSmith; @HSS].  Let $Q$ be the complete quiver of sections for $\mathcal{L} = (\mathscr{O}_X, L_1,\dotsc,
  L_r)$. Since $L_r$ is very ample and $L_0, \dotsc, L_{r-1}$ are basepoint-free, it follows that the line bundle $L :=
  \bigotimes_{i\in Q_0} L_i$ is very ample.  Since Theorem 2.1 in [@HSS] proves that $H^0(X, L_0) \otimes_{\ensuremath{\Bbbk}}\dotsb \otimes_{\ensuremath{\Bbbk}}H^0(X,L_r) \rTo H^0(X, L)$ is surjective, Corollary \[cor:veryample\] completes the proof. 

\[thm:ample\] If $Q$ is a very ample quiver of sections, then we can recover the line bundles $L_0, \dotsc, L_r$ as the restriction of the tautological line bundles on $Y = |\mathcal{L}|$.

If we identify $\operatorname{Wt}(Q)$ with ${\ensuremath{\mathbb{Z}}}^r$ by choosing the weights $(\chi_1 - \chi_0, \dotsc, \chi_r - \chi_0)$ as an ordered basis, then the projection map ${\ensuremath{\mathbb{Z}}}^{Q_0} \rTo {\ensuremath{\mathbb{Z}}}^r$ induces an isomorphism between $G$ and the subgroup $\{ (g_0, \dotsc, g_r) \in
  ({\ensuremath{\Bbbk}}^*)^{r+1} : g_0 = 1 \}$ of $({\ensuremath{\Bbbk}}^*)^{Q_0}$. This isomorphism determines a $G$-equivariant vector bundle $\bigoplus_{i \in Q_0}
  \mathscr{O}_{{\ensuremath{\mathbb{A}}}^{Q_1}}$; specifically, the $i$-th component corresponds to the $S_Y$-module $S_Y(\chi_i - \chi_0)$, where $\bigl( S_Y(\theta') \bigr)_\theta = (S_Y)_{\theta' + \theta}$. If follows that the tautological line bundles on $Y$ are $\mathscr{O}_Y, \mathscr{O}_Y(\chi_1 - \chi_0), \dotsc,
  \mathscr{O}_Y(\chi_r - \chi_0)$. Restricting to ${\ensuremath{\mathbb{V}}}(I_Q)$, we obtain a $G$-equivariant vector bundle $\bigoplus_{i \in Q_0}
  \mathscr{O}_{{\ensuremath{\mathbb{V}}}(I_Q)}$ where the $i$-th component corresponds to $\bigl( S_Y/ I_Q \bigr)(\chi_i - \chi_0)$. Since $\varphi_Q \colon
  X \rTo Y$ is a closed embedding, Proposition \[pro:ample\] implies that $\bigl( (S_Y/I_Q)[y^{- \widehat\sigma}] \bigr)_{[0]} \cong
  \bigl( S_X[x^{- \widehat\sigma}] \bigr)_{[0]}$ for all $\sigma \in
  \Sigma_X(d)$. Hence, the $S_Y$-module $\bigl( S_Y/ I_Q
  \bigr)(\chi_i - \chi_0)$ corresponds to $\operatorname{pic}(\chi_i - \chi_0) =
  L_i$ on $X \cong \varphi_Q(X)$.

If we identify $\operatorname{Wt}(Q)$ with ${\ensuremath{\mathbb{Z}}}^r$ by choosing $(\chi_0 - \chi_1,
  \dotsc, \chi_0 - \chi_r)$ as an ordered basis then the restriction of the tautological line bundles would yield the inverse line bundles $\mathcal{O}_X, L_1^{-1}, \dotsc, L_r^{-1}$.

Representations of Bound Quivers {#sec:boundreps}
================================

This section connects the ideal of relations on a quiver of sections to the geometry of its multilinear series.  Throughout this section, let $(Q,R)$ be a complete, bound quiver of sections for $\mathcal{L} =
(\mathscr{O}_X,L_1, \dotsc, L_r)$ of line bundles on $X$ and let $Y =
|\mathcal{L}|$ be the multilinear series of $Q$.

The ideal of relations defines an algebraic subset of ${\ensuremath{\mathbb{A}}}^{Q_1}$. More precisely, the map sending the path $p = a_1 \dotsb a_\ell$ in $Q$ to the monomial $y_{a_1} \dotsb y_{a_\ell} \in S_Y$ extends to give a ${\ensuremath{\Bbbk}}$-linear map from ${\ensuremath{\Bbbk}}Q$ into $S_Y$. Let $I_R$ be the ideal in $S_Y$ generated by the image of $R$ under this map; it is a binomial ideal because $R$ is spanned by differences $p - p' \in {\ensuremath{\Bbbk}}Q$. Since these differences satisfy $\operatorname{tl}(p) = \operatorname{tl}(p')$, $\operatorname{hd}(p)
= \operatorname{hd}(p')$, and ${\operatorname{div}}(p) = {\operatorname{div}}(p')$, $I_R$ is homogeneous with respect to the $\operatorname{Wt}(Q)$-grading on $S_Y$ and is contained in $I_Q$.

The following examples illustrate various possible relations between $I_R$ and $I_Q$.

\[exa:PP2O2\] Let $X = {\ensuremath{\mathbb{P}}}^1$ and let $\mathcal{L} = \bigl( \mathscr{O}_X,
  \mathscr{O}_X(2) \bigr)$; Example \[exa:2ptY\] describes $|\mathcal{L}|$. There are no paths of length greater than $1$ in $Q$, so $R$ and $I_R$ are both the zero ideal.  Since $I_Q$ is the toric ideal associated to the matrix

$$\left[ \text{\footnotesize $\begin{array}{rrr}
      -1 & -1 & -1 \\ 
      1 &  1 &  1 \\
      2 &  1 &  0 \\
      0 &  1 &  2
    \end{array}$} \right] \, ,$$ we have $I_Q = ( y_0y_2 - y_1^2)$. Thus, ${\ensuremath{\mathbb{V}}}(I_Q)$ is a closed subvariety of ${\ensuremath{\mathbb{V}}}(I_R) = {\ensuremath{\mathbb{A}}}^{3} = {\ensuremath{\mathbb{A}}}^{Q_1}$.

\[exa:FF1Z\] Let $X = {\ensuremath{\mathbb{F}}}_1$ and $\mathcal{L} = \bigl( \mathscr{O}_X,
  \mathscr{O}_{X}(1,0), \mathscr{O}_{X}(0,1) \bigr)$. Since Example \[exa:FF1\] shows that $R = (0)$ and Example \[exa:FF2image\] shows that $I_Q = (0)$, it follows that $I_R = I_Q = (0)$; in other words, ${\ensuremath{\mathbb{V}}}(I_Q) = {\ensuremath{\mathbb{V}}}(I_R) =
  {\ensuremath{\mathbb{A}}}^4={\ensuremath{\mathbb{A}}}^{Q_1}$.

\[exa:FF2Z\] Let $X = {\ensuremath{\mathbb{F}}}_2$ and $\mathcal{L} = \bigl( \mathscr{O}_X,
  \mathscr{O}_{X}(1,0), \mathscr{O}_{X}(0,1), \mathscr{O}_{X}(1,1)
  \bigr)$. Using the description of $R$ in Example \[exa:FF2\], we see that $$\begin{aligned}
    I_R &= (y_2y_4-y_1y_5, y_4y_8-y_5y_7, y_2y_6-y_3y_8, y_1y_6 -
    y_3y_7) \\ &= (y_2y_4-y_1y_5, y_1y_6-y_3y_7, y_2y_6-y_3y_8,
    y_2y_7-y_1y_8, y_5y_7-y_4y_8) \cap (y_3, y_4, y_5, y_6) \, . 
  \end{aligned}$$ The description of $I_Q$ given in Example \[exa:FF2image\] implies that $I_Q$ is a primary component of $I_R$. Geometrically, ${\ensuremath{\mathbb{V}}}(I_Q)$ is the unique component of ${\ensuremath{\mathbb{V}}}(I_R)$ not lying in a linear subspace. 

Let $W = (W_i, w_a)$ be a representation of $Q$. For any nontrivial path $p = a_1 \dotsb a_\ell$, the evaluation of $W$ on $p$ is the ${\ensuremath{\Bbbk}}$-linear map $w_p \colon W_{\operatorname{tl}(p)} \rTo W_{\operatorname{hd}(p)}$ defined by the composition $w_p = w_{a_1} \dotsb w_{a_\ell}$. This definition extends to ${\ensuremath{\Bbbk}}$-linear combinations of paths with a common head and a common tail.  A representation of the bound quiver $(Q,R)$ is a representation $W$ of $Q$ such that $w_p - w_{p'} = 0$ for all $p-p'
\in R$. Consequently, a point in the representation space ${\ensuremath{\mathbb{A}}}^{Q_1}$ for $Q$ corresponds to a representation for $(Q,R)$ if and only it lies in the subscheme ${\ensuremath{\mathbb{V}}}(I_R)$. The category of representations of the bound quiver $(Q,R)$ of dimension vector $\sum_{i\in Q_0}
\chi_i\in {\ensuremath{\mathbb{N}}}^{Q_0}$ is equivalent to the category of $({\ensuremath{\Bbbk}}Q/R)$-modules that are isomorphic as $\bigl( \bigoplus_{i\in Q_0} {\ensuremath{\Bbbk}}e_i \bigr)$-modules to $\bigoplus_{i\in Q_0} {\ensuremath{\Bbbk}}e_i$.

The ideal $I_R$ is homogeneous with respect to the $\operatorname{Wt}(Q)$-grading on $S_Y$, so the subscheme ${\ensuremath{\mathbb{V}}}(I_R)$ is $G$-invariant where $G =
\operatorname{Hom}_{\ensuremath{\mathbb{Z}}}\bigl( \operatorname{Wt}(Q), {\ensuremath{\Bbbk}}^* \bigr)$. The GIT-chamber decomposition of $\operatorname{Wt}(Q) \otimes_{\ensuremath{\mathbb{Z}}}{\ensuremath{\mathbb{Q}}}$ arising from the $G$-action on ${\ensuremath{\mathbb{V}}}(I_R)$ coarsens that for the $G$-action on $\mathbb{A}^{Q_1}$; see Remark \[rem:chamber\].  Let $\Theta$ denote the GIT-chamber arising from the $G$-action on ${\ensuremath{\mathbb{V}}}(I_R)$ containing $\vartheta = \sum_{i\in
  Q_0} (\chi_i-\chi_0)$. Proposition 5.3 in [@King1] shows that, for $\theta \in \Theta$, the GIT-quotient $$\mathcal{M}_\theta(Q,R) := {\ensuremath{\mathbb{V}}}(I_R) {\ensuremath{/\!\!/_{\!\theta}\,}}G = \operatorname{Proj}\Bigl(
\bigoplus\limits_{k \in {\ensuremath{\mathbb{N}}}} \bigl( \tfrac{S_Y}{I_R} \bigr)_{k
  \theta} \Bigr)$$ is the fine moduli space for $\theta$-stable representations of $(Q,R)$. Equivalently, if $\bigoplus_{i \in Q_0} {\ensuremath{\Bbbk}}e_i$ denotes the subalgebra of $\operatorname{End}\bigl( \bigoplus_{i \in Q_0} L_i \bigr)$ generated by the primitive orthogonal idempotents $e_i$ for $i \in Q_0$, then Proposition \[pro:algebra\] implies that $\mathcal{M}_\theta(Q,R)$ is the fine moduli space of $\theta$-stable $\operatorname{End}\bigl( \bigoplus_{i
  \in Q_0} L_i \bigr)$-modules that are isomorphic as $\bigl(
\bigoplus_{i \in Q_0} {\ensuremath{\Bbbk}}e_i \bigr)$-modules to $\bigoplus_{i \in Q_0}
  {\ensuremath{\Bbbk}}e_i$.

\[thm:finequotient\] If $Q$ is a very ample quiver of sections, then the following are equivalent:

1.   the ideal $I_Q$ equals ideal quotient $(I_R : B_Y^\infty)$;

2.   for all $\theta \in \Theta$, the map $\varphi_Q$ induces an isomorphism from $X$ to $\mathcal{M}_{\theta}(Q,R)$.

The equivalence of (b) and (c) in Proposition \[pro:smooth\] implies that the moduli space $\mathcal{M}_\theta(Q,R) = {\ensuremath{\mathbb{V}}}(I_R)
  {\ensuremath{/\!\!/_{\!\theta}\,}}G$ is the geometric quotient of ${\ensuremath{\mathbb{V}}}(I_R) \setminus {\ensuremath{\mathbb{V}}}(B_Y)$ by the group $G$. Since $Q$ is a very ample quiver of sections, Proposition \[pro:ample\] proves that the map $\varphi_Q$ induces an isomorphism from $X$ to $\varphi_Q(X)$ and Proposition \[pro:ideal\] establishes that $\varphi_Q(X)$ is the geometric quotient of ${\ensuremath{\mathbb{V}}}(I_Q) \setminus {\ensuremath{\mathbb{V}}}(B_Y)$ by $G$. Since $I_Q$ is prime, the locally closed subscheme ${\ensuremath{\mathbb{V}}}(I_Q) \setminus
  {\ensuremath{\mathbb{V}}}(B_Y)$ equals ${\ensuremath{\mathbb{V}}}(I_R) \setminus {\ensuremath{\mathbb{V}}}(B_Y)$ if and only if we have $I_Q = I_R : B_Y^\infty$.

A list $\mathcal{L} = (\mathscr{O}_X, L_1, \dotsc, L_r)$ of line bundles on $X$ is *fine* if the complete bound quiver of sections for $\mathcal{L}$ is very ample and satisfies either of the equivalent conditions in Theorem \[thm:finequotient\].  The next result shows that every projective toric variety has many fine lists. 

\[thm:fine\] Let $L_1, \dotsc, L_{r-2}$ be basepoint-free line bundles on $X$. If the subsemigroup of $\operatorname{Pic}(X)$ generated by $L_1, \dotsc, L_{r-2}$ contains an ample line bundle, then there exist line bundles $L_{r-1}$ and $L_r$ such that the list $\mathcal{L} =
  (\mathscr{O}_X, L_1, \dotsc, L_r)$ is fine. 

We divide the proof into three parts. 

This part is similar to Proposition \[pro:ample\].  By choosing sufficiently large positive integers $b_1, \dotsc, b_{r-2}$, we may assume that the line bundle $L_{r-1} := L_1^{b_1} \otimes
    \dotsb \otimes L_{r-2}^{b_{r-1}}$ is $\mathscr{O}_X$-regular with respect to $L_1, \dotsc, L_{r-2}$. Set $L_r := L_{r-1}^2$. By increasing the $b_i$ if necessary, we may also assume that $L_r$ is very ample.  Let $Q$ be the complete quiver of sections for $\mathcal{L} = (\mathscr{O}_X, L_1, \dotsc, L_r)$. Since $L_r$ is very ample and $L_1, \dotsc, L_{r-1}$ are basepoint-free, it follows that $L := \bigotimes_{i \in Q_0} L_i$ is very ample.  Since Theorem 2.1 in [@HSS] implies that $H^0(X,L_1)
    \otimes_{\ensuremath{\Bbbk}}\dotsb \otimes_{\ensuremath{\Bbbk}}H^0(X,L_r) \rTo H^0(X,L)$ is surjective, Corollary \[cor:veryample\] implies that $Q$ is very ample. 

By definition, $I_Q$ is the toric ideal associated to the map $\pi
    \colon {\ensuremath{\mathbb{N}}}^{Q_1} \rTo \operatorname{Wt}(Q) \oplus \operatorname{CDiv}(X)$. It suffices by Lemma 12.2 in [@Sturmfels] to construct a subset $\mathcal{C}$ that generates the kernel of $\pi$ as an abelian group and satisfies $(y^{v_+} - y^{v_-} : v_+- v_- = v \in \mathcal{C})
    \subseteq I_R$. Since $L_{r-1}$ is $\mathscr{O}_X$-regular and the $b_i$ are positive, Theorem 2.1 in [@HSS] shows that $H^0(X,L_{r-1}^{\,} \otimes
    L_i^{-1}) \otimes_{\ensuremath{\Bbbk}}H^0(X,L_{r-1}) \rTo H^0(X,L_r^{\,} \otimes
    L_i^{-1})$ is surjective for all $1 \leq i \leq r-1$. Hence, every path in $Q$ from $0$ to $r$ passes through $r-1 \in Q_0$. Moreover, because $Q$ is complete, the set $\mathcal{A}$ of the arrows from $r-1$ to $r$ in $Q$ corresponds to the set of nonzero $T_X$-invariant elements in $H^0(X,L_{r-1})$. Since the set $\mathcal{P}$ of paths from $0$ to $r-1$ in $Q$ are labelled by nonzero $T_X$-invariant elements in $H^0(X,L_{r-1})$, there is a surjective function $\Psi \colon \mathcal{P} \rTo \mathcal{A}$ such that ${\operatorname{div}}(p) = {\operatorname{div}}\bigl( \Psi(p) \bigr)$ for all $p \in
    \mathcal{P}$. For $(a, a', p, p') \in \mathcal{A}^2 \times
    \mathcal{P}^2$, we have ${\operatorname{div}}(p) + {\operatorname{div}}(a) = {\operatorname{div}}(p') + {\operatorname{div}}(a)$ if and only if $p a - p' a' \in R$; set $$\mathcal{C} := \{ f(p) + f(a) - f(p') - f(a') : pa-p'a' \in R \}
    \subseteq \operatorname{Cir}(Q) \, ,$$ where $f(\gamma)$ is the element in ${\ensuremath{\mathbb{Z}}}^{Q_1}$ associated to a walk $\gamma$ in $Q$ defined in §\[sub:quivers\]. 

To analyze $\mathcal{C}$, we use an “elongation” operation on circuits in $Q$. Because $Q$ is acyclic, we may assume every circuit $\gamma = \alpha_1^{\,} \alpha_2^{-1} \alpha_3^{\,} \dotsb
    \alpha_{2\ell -1}^{\,} \alpha_{2 \ell}^{-1}$ is an alternating sequence of forward paths $\alpha_1, \alpha_3, \dotsc,
    \alpha_{2\ell -1}$ and backward paths $\alpha_2^{-1},
    \alpha_4^{-1}, \dotsc, \alpha_{2 \ell}^{-1}$. Since $Q$ is connected, there exists at least one path from the unique source $0$ to each $i \in Q_0$. Similarly, the choice of $L_r$ implies that there is at least one path from each $i \in Q_0$ to $r$. Let $\widehat{\gamma}$ denote a closed walk obtained from the circuit $\gamma$ via the following procedure: for $i=1,3, \dotsc,
    2\ell-1$, choose a path $\beta_i$ from $\operatorname{hd}(\alpha_i)$ to $r$; for $i = 2, 4, \dotsc, 2\ell$, choose a path $\beta_i$ from $0$ to $\operatorname{tl}(\alpha_i)$; let $\beta_{0} = \beta_{2 \ell}$ and set $\widehat{\gamma} := \beta_{0}^{\,} \alpha_1^{\,}\beta_1^{\,}
    \beta_1^{-1}\alpha_2^{-1} \beta_{2}^{-1}\beta_2^{\,} \alpha_3^{\,}
    \beta_3^{\,} \beta_3^{-1} \dotsb \alpha_{2 \ell}^{-1}
    \beta_{2\ell}^{-1}$. Observe that $\widehat\gamma =
    \widehat{p}_1^{\,} \widehat{p}_2^{-1} \widehat{p}_3^{\,} \dotsb
    \widehat{p}_{2\ell -1}^{\,} \widehat{p}_{2 \ell}^{-1}$ is an alternating sequence of forward and backward paths between $0$ and $r$, where $\widehat{p}_i = \beta_{i-1}\alpha_{i}\beta_{i}$ for odd $i$, $\widehat{p}_i^{-1} =
    \beta_{i-1}^{-1}\alpha_i^{-1}\beta_{i}^{-1}$ for even $i$. Thus, we have ${\operatorname{div}}(\gamma) = {\operatorname{div}}(\widehat\gamma) \in \operatorname{CDiv}(X)$ and $$f(\gamma) = f(\widehat\gamma) = \sum\limits_{i=0}^{\ell} \bigl(
    f(\widehat{p}_{2i-1}) \bigr) - \sum\limits_{i=0}^{\ell} \bigl(
    f(\widehat{p}_{2i}) \bigr) \in \operatorname{Cir}(Q)$$ where $f(\widehat{p}_i) \in {\ensuremath{\mathbb{N}}}^{Q_1}$.

To see that $\mathcal{C}$ spans the lattice $\operatorname{Ker}(\pi)$, fix $u
    \in \operatorname{Ker}(\pi)$. Since $\pi = (\operatorname{inc}, {\operatorname{div}})$, the exact sequence implies that $u \in \operatorname{Cir}(Q)$. Theorem 5.2 in [@Biggs] shows that the circulation lattice is generated by the circuits, so $u = \sum_{i} f(\gamma_i)$ where $\gamma_i$ is a circuit in $Q$. By using the elongation operation and regrouping the sum, we have $u = \sum_{i} \bigl( f(\widehat{p}_{2i-1}) -
    f(\widehat{p}_{2i}) \bigr)$ where each $\widehat{p}_i$ is a path from $0$ to $r$. Since each path in $Q$ from $0$ to $r$ passes through $r-1$, it follows that $\widehat{p}_i = p_i a_i$ where $p_i \in \mathcal{P}$ and $a_i \in \mathcal{A}$. Hence, we have $u = \textstyle\sum\nolimits_{i} \bigl( f(p_{2i-1}) + f(a_{2i-1})
    - f(p_{2i}) - f(a_{2i}) \bigr)$. We decompose this expression into a sum of elements in $\mathcal{C}$ by exploiting properties of the sets $\mathcal{A}$ and $\mathcal{P}$ arising from our choice of $L_{r-1}$. To be specific, let $I'$ be the toric ideal associated to the map ${\ensuremath{\mathbb{N}}}^{\mathcal{A}} \rTo \operatorname{CDiv}(X)$ given by $\chi_{a} \mapsto {\operatorname{div}}(a)$. The identification of $\mathcal{A}$ with the nonzero $T_X$-invariant elements in $H^0(X,L_{r-1})$ implies that $I'$ is the ideal of $\varphi_{|L_{r-1}|}(X)$. Since $L_{r-1}$ is $\mathscr{O}_X$-regular, Theorem 1.1 in [@HSS] establishes that $I'$ is generated by quadrics.  Hence, if ${\ensuremath{\Bbbk}}[z_a : a \in \mathcal{A}] = {\ensuremath{\Bbbk}}[{\ensuremath{\mathbb{N}}}^{\mathcal{A}}]$ and $$\mathcal{R} := \{ (a_0,a_1,a_2,a_3) \in \mathcal{A}^4 : {\operatorname{div}}(a_0)
    + {\operatorname{div}}(a_1) = {\operatorname{div}}(a_2) + {\operatorname{div}}(a_3) \} \, ,$$ then we have $I' = \, \bigr( z_{a_0}z_{a_1} - z_{a_2}z_{a_3} :
    (a_0,a_1,a_2,a_3) \in \mathcal{R} \bigr)$. For $1 \leq i \leq
    \ell$, we define $c_i := f(p_{2i-1}) + f(\Psi(p_{2i})) - f(p_{2i})
    - f(\Psi(p_{2i-1}))$. Since ${\operatorname{div}}(p_i) = {\operatorname{div}}\bigl( \Psi(p_i)
    \bigr)$ for $1 \leq i \leq \ell$, each $c_i$ belongs to $\mathcal{C}$ and we have $$\begin{aligned}
      u &= \sum\limits_i \bigl( f(p_{2i-1}) + f(a_{2i-1}) - f(p_{2i})
      - f(a_{2i}) \bigr) \\ &= \sum\limits_i \bigl( f(\Psi(p_{2i-1}))
      + f(a_{2i-1}) - f(\Psi(p_{2i})) - f(a_{2i}) \bigr) +
      \sum\limits_i c_i \, . 
    \end{aligned}$$ Given $v \in {\ensuremath{\mathbb{Z}}}^{\mathcal{A}}$ satisfying ${\operatorname{div}}(v) = 0$, Theorem 5.3 in [@Sturmfels] applied to the generators of the toric ideal $I'$ yields $$\begin{aligned}
      v = \!\!  \sum\limits_{(a_0,a_1,a_2,a_3) \in \mathcal{R}'} \!\! ( 
      \chi_{a_0} + \chi_{a_1} - \chi_{a_2} - \chi_{a_3} ) = \!\! 
      \sum\limits_{(a_0,a_1,a_2,a_3) \in \mathcal{R}'} \!\!  \bigl(
      f(a_0) + f(a_1) - f(a_2) - f(a_3) \bigr) \, ,
    \end{aligned}$$ where $\mathcal{R}'$ is a multiset of elements from $\mathcal{R}$. Applying this to the difference $u - \sum_i c_i$ implies that $u =
    \textstyle\sum\nolimits_{(a_0,a_1,a_2,a_3) \in \mathcal{R}'} \bigl(
    f(a_0) + f(a_1) - f(a_2) - f(a_3) \bigr) + \textstyle\sum\nolimits_i
    c_i$ where $\mathcal{R}'$ is some multiset of elements from $\mathcal{R}$. For $1 \leq i \leq \ell$, set $c_i' :=
    f(\widetilde{a}_0) + f(a_2) - f(\widetilde{a}_2) - f(a_0)$ where $\widetilde{a}_j \in \mathcal{P}$ satisfies $\Psi(\widetilde{a}_j)
    = a_j$ for $j = 0, 2$. Hence, we obtain $$\begin{aligned}
      u &= \sum\limits_{(a_0,a_1,a_2,a_3) \in \mathcal{R}'} \bigl(
      f(\widetilde{a}_0) + f(a_1) - f(\widetilde{a}_2) - f(a_3) \bigr)
      - \sum\limits_i c_i' + \sum\limits_i c_i
    \end{aligned}$$ where $\mathcal{R}'$ is a multiset of elements from $\mathcal{R}$. Since each summand belongs to $\mathcal{C}$, we see that $u$ lies in the lattice spanned by $\mathcal{C}$ and $I_Q = \bigl( I_R :
    (\prod_{a \in Q_1} y_a)^\infty \bigr)$.

For $u \in {\ensuremath{\mathbb{Z}}}^{Q_1}$, set $y^u := \prod_{a \in Q_1} y_a^{u_a}$. For any subset $Q_1' \subseteq Q_1$, let $f(Q_1') := \sum_{a \in
      Q_1'} \chi_a$ so that $y^{f(Q_1')} := \prod_{a \in Q_1'} y_a$. The definition of $B_Y$ given in Proposition \[pro:smooth\] (b) implies that $(I_R : B_Y^\infty) = \bigcap_{Q'} \bigl( I_R :
    (y^{f(Q'_1)} )^\infty \bigr)$ where the intersection is over all spanning tree $Q' \subseteq Q$ rooted at $0$. Because $\bigl( I_R
    : ( y^{f(Q_1')})^\infty \bigr)$ is a subset of $\bigl( I_R : (
    \prod_{a \in Q_1} y_a )^\infty \bigr)$, it is enough to show that each spanning tree $Q' \subseteq Q$ rooted at $0$ satisfies $\bigl( I_R : ( y^{f(Q_1')} )^\infty \bigr) = \bigl( I_R :
    (\prod_{a \in Q_1} y_a )^\infty \bigr)$.

The technique of proof follows Example 2.3 in [@BLG].  By increasing the $b_i$ from part 1 if necessary, we may assume that there exists $s \in H^0(X,L_{r-1})$ such that the corresponding lattice point in the polytope $P$ associated $L_{r-1}$ lies in the interior.  Fix a spanning tree $Q'$ rooted at $0$. Let $a_s$ be the unique arrow $a_s \in \mathcal{A} \cap Q_1'$ and let $p_s$ be a path in $Q$ satisfying ${\operatorname{div}}(p_s) = {\operatorname{div}}(s) = {\operatorname{div}}(a_s)$. Observe that $y_{a_s}$ is invertible in $S_Y[y^{-f(Q_1')}]$. For any path $p \in \mathcal{P}$, we have $pa_s - p_s \Psi(p) \in R$, so $y^{f(p)} - y^{f(p_s)}y_{\Psi(p)}y_{a_s}^{-1}$ belongs to $I_R
    S_Y[y^{-f(Q_1')}]$. Hence, for any $p,p' \in \mathcal{P}$ and $a,
    a' \in \mathcal{A}$, $$\begin{aligned}
      y^{f(p)}y_a - y^{f(p')}y_{a'} &=
      y^{f(p_s)}y_{\Psi(p)}y_ay_{a_s}^{-1} -
      y^{f(p_s)}y_{\Psi(p')}y_{a'}y_{a_s}^{-1} =
      y^{f(p_s)}y_{a_s}^{-1} ( y_{\Psi(p)} y_a - y_{\Psi(p')} y_{a'} )
    \end{aligned}$$ in $S_Y[y^{- f(Q_1')}]/ I_R S_Y[y^{-f(Q_1')}]$. Thus, if $$J := I_R + \bigl( y_{a_0}y_{a_1} - y_{a_2}y_{a_3} :
    (a_0,a_1,a_2,a_4) \in \mathcal{R} \bigr) \, ,$$ then we have $J S_Y[y^{- f(Q_1')}] = I_R S_Y[y^{- f(Q_1')}]$.

By assumption, $L_{r-1}$ is ample, so the vertices of the polytope $P$ are lattice points.  Since the lattice point corresponding to $s$ lies in the interior of $P$, we can express $s$ in the form $\tfrac{1}{k}\sum_j c_j s_j$ where $k$, $c_j$ are positive integers and the $s_j$ correspond to the vertices of $P$. It follows that $z_{a_s}^k - \prod_j z_{a_{s_j}}^{c_j}$ lies in the toric ideal $$I' = \bigl( z_{a_0}z_{a_1} - z_{a_2}z_{a_3} : (a_0,a_1,a_2,a_3)\in
    \mathcal{R} \big)$$ introduced in part 2.  Changing variables from $z$’s to $y$’s, we see that $y_{a_s}^k - \prod_j y_{a_{s_j}}^{c_j} \in J$. Since $y_{a_s}$ is invertible in $S_Y[y^{-f(Q_1')}]$, it now follows that all the $y_{a_{s_j}}$ are invertible in the quotient ring $S_Y[y^{- f(Q_1')}] / I_R S_Y[y^{- f(Q_1')}]$. Moreover, we see that $y_a$ is invertible in $S_Y[y^{- f(Q_1')}] / I_R S_Y[y^{-
      f(Q_1')}]$ for all $a \in \mathcal{A}$ because the lattice point of $P$ corresponding to $a \in \mathcal{A}$ is a positive rational combination of some vertices and each variable $y_{a_{s_j}}$ that corresponds to a vertex is invertible in $S_Y[y^{- f(Q_1')}] / I_R
    S_Y[y^{- f(Q_1')}]$.

If $p' \in \mathcal{P}$ is the unique path supported on the set $Q_1'$, then $y^{f(p')}$ is invertible in $S_Y[y^{-f(Q_1')}] / I_R
    S_Y[y^{- f(Q_1')}]$. Let $a' \in \mathcal{A}$ is the unique arrow satisfying ${\operatorname{div}}(p') = {\operatorname{div}}(a')$. The previous paragraph shows that $y_{a'}$ in invertible in $S_Y[y^{-f(Q_1')}] / I_R S_Y[y^{-
      f(Q_1')}]$. For any path $p \in \mathcal{P}$, we have $pa' - p'
    \Psi(p) \in R$ and the identity $y^{f(p)} =
    y^{f(p')}y_{\Psi(p)}y_{a'}^{-1}$ in $I_R S_Y[y^{- f(Q_1')}]$. Each monomial on the right side of this identity is invertible in $S_Y[y^{- f(Q_1')}] / I_R S_Y[y^{- f(Q_1')}]$ which implies that every variable that divides $y^{f(p)}$ is also invertible.  Since the path $p \in \mathcal{P}$ was arbitrary, we conclude that, for all $a \in Q_1$, $y_a$ is invertible in $S_Y[y^{- f(Q_1')}] / I_R
    S_Y[y^{- f(Q_1')}]$. Therefore, Corollary 2.6 in [@BLG] implies that $\bigl(I_R : ( y^{f(Q_1')} )^\infty \bigr) = \bigl(
    I_R : (\prod_{a \in Q_1} y_a )^\infty \bigr)$.

Since part (1) shows that $Q$ is very ample, and combining parts (2) and (3) proves that $Q$ satisfies condition (a) in Theorem \[thm:finequotient\], we conclude that $\mathcal{L}$ is fine. 

By combining Theorem \[thm:fine\] and Theorem \[thm:finequotient\], it follows that there are many list $\mathcal{L} = (\mathcal{O}_X, L_1, \dotsc, L_r)$ of line bundles on $X$ such that the induced morphism $\varphi_{|\mathcal{L}|} \colon X
  \rTo |\mathcal{L}|$ identifies $X$ with the fine moduli space $\mathcal{M}_{\vartheta}(Q,R)$. Theorem \[thm:ample\] implies that the tautological bundles on $\mathcal{M}_{\vartheta}(Q,R)$ coincide with the line bundles $\mathcal{O}_X, L_1, \dotsc, L_r$.

A priori, $I_Q$ depends on the divisors labelling the arrows in $Q$. However, if $\mathcal{L}$ is fine then $I_Q$ depends only on $I_R$ and hence only the bound quiver $(Q,R)$.

For $X = {\ensuremath{\mathbb{P}}}^1$, Example \[exa:PP2O2\] shows that $\mathcal{L} =
  \bigl( \mathscr{O}_X, \mathscr{O}_X(2) \bigr)$ is not fine.  To obtain a fine list, we add an appropriate line bundle: $\mathcal{L}'
  := \bigl( \mathscr{O}_X, \mathscr{O}_X(2), \mathscr{O}_X(4) \bigr)$. The complete quiver of sections for $\mathcal{L}'$ appears in Figure \[fig:simplestfine\]. 

Corollary \[cor:veryample\] implies that $|\mathcal{L}'|$ is very ample.  Since $$\begin{aligned}
    I_Q &= (y_2^2-y_1y_3,y_5^2-y_4y_6, y_3y_5-y_2y_6, y_2y_5-y_1y_6,
    y_3y_4-y_1y_6,y_2y_4-y_1y_5) \;\; \text{and} \\
    I_R &= (y_3y_5 - y_2y_6, y_3y_4-y_2y_5, y_2y_5-y_1y_6,
    y_2y_4-y_1y_5) 
    = I_Q \cap (y_1,y_2,y_3)\cap (y_4,y_5,y_6) \, ,
  \end{aligned}$$ it follows that $\mathcal{L}$ is fine. 

\[ex:3foldflop\] Let $X$ be the smooth toric threefold determined by the following fan $\Sigma_X$ in ${\ensuremath{\mathbb{R}}}^3$: the rays $\Sigma_X(1)$ are generated by the vectors $v_1 := (1,0,0)$, $v_2 := (0,1,0)$, $v_3 := (-1,-1,-1)$, $v_4 := (0,1,1)$, $v_5 := (1,0,1)$ and the minimal nonfaces correspond to $\{ v_1, v_3, v_4 \}$ and $\{ v_2, v_5 \}$. The induced triangulation of the $2$-sphere is given in Figure \[fig:3fold\] (a).  There is a flop $\begin{diagram}[inline,size=1.9em,l=>1.9em,tight] X & \rDashto &
    X' \end{diagram}$ where the toric variety $X'$ is the determined by the triangulation of $\Sigma_X(1)$ with minimal nonfaces $\{ v_1,
  v_4\}$ and $\{ v_2, v_3, v_5 \}$. For $(k, \ell) \in {\ensuremath{\mathbb{Z}}}^2$, write $\mathscr{O}_{X}(k,\ell) := \mathscr{O}_{X}(kD_3+\ell D_2) \in
  \operatorname{Pic}(X)$. The complete quiver of sections for $\mathcal{L} = \bigl(
  \mathscr{O}_{X}, \mathscr{O}_{X}(0,1), \mathscr{O}_{X}(1,0),
  \mathscr{O}_{X}(1,1), \mathscr{O}_{X}(2,0), \mathscr{O}_{X}(2,1)
  \bigr)$ appears in Figure \[fig:3fold\] (b). 

Since we have $$\begin{aligned}
    I_Q &= \left(
      \begin{array}{c}
        y_9y_{13}-y_8y_{14}, y_2y_{13}-y_1y_{14}, y_9y_{12}-y_7y_{14},
        y_8y_{12}-y_7y_{13}, y_5y_{10}-y_4y_{11},\\  y_2y_8-y_1y_9,
        y_5y_7-y_6y_{11}, y_4y_7-y_6y_{10}, y_2y_6-y_3y_9,
        y_1y_6-y_3y_8, \\ y_2y_5y_{12}-y_3y_{11}y_{14}, 
        y_1y_5y_{12}-y_3y_{11}y_{13}, y_2y_4y_{12}-y_3y_{10}y_{14},
        y_1y_4y_{12}-y_3y_{10}y_{13}
      \end{array}
    \right) \\
    I_R &= \left(
      \begin{array}{c} 
        y_9y_{12}-y_7y_{14}, y_8y_{12}-y_7y_{13},
        y_5y_7-y_6y_{11}, y_4y_7-y_6y_{10}, y_2y_6-y_3y_9, \\
        y_1y_6-y_3y_8,   y_9y_{11}y_{13}-y_8y_{11}y_{14},
        y_9y_{10}y_{13}-y_8y_{10}y_{14}, y_5y_9y_{10}-y_4y_9y_{11}, \\
        y_5y_8y_{10}-y_4y_8y_{11}, y_2y_5y_8-y_1y_5y_9,
        y_2y_4y_8-y_1y_4y_9
      \end{array}
    \right) \\
    &= I_Q \cap ( y_{12}, y_{11}, y_{10}, y_7, y_2y_8-y_1y_9,
    y_2y_6-y_3y_9, y_1y_6-y_3y_8 ) \\ &\qquad \; \cap ( y_6, y_5, y_4,
    y_3, y_9 y_{13}-y_8y_{14}, y_9y_{12}-y_7y_{14},y_8y_{12}-y_7y_{13}
    ) \\ &\qquad \; \cap ( y_{12}, y_{11}, y_{10}, y_7, y_6, y_5, y_4,
    y_3 ) \cap ( y_9, y_8, y_7, y_6 )  \, ,
  \end{aligned}$$ it follows that $\mathcal{L}$ fine.  Observe that ${\ensuremath{\mathbb{V}}}(I_Q)$ has four components contained in coordinate hyperplanes of ${\ensuremath{\mathbb{A}}}^{Q_1} =
  {\ensuremath{\mathbb{A}}}^{14}$ and the points in these components correspond to representations of disconnected subquivers of $Q$.
