TY - JOUR

T1 - Projective toric varieties as fine moduli spaces of quiver representations

AU - Craw, Alastair

AU - G. Smith, Gregory

PY - 2008/12

Y1 - 2008/12

N2 - 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 $\bigl(\bigoplus_{i=0}^{r} L_i \bigr)$ where $\mathcal{L} := (\mathscr{O}_X,L_1, ...c, L_r)$ 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 $X \to |\mathcal{L}|$. 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 $(Q,R)$.

AB - 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 $\bigl(\bigoplus_{i=0}^{r} L_i \bigr)$ where $\mathcal{L} := (\mathscr{O}_X,L_1, ...c, L_r)$ 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 $X \to |\mathcal{L}|$. 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 $(Q,R)$.

UR - http://www.scopus.com/inward/record.url?scp=58049157396&partnerID=8YFLogxK

UR - http://dx.doi.org/10.1353/ajm.0.0027

UR - http://arxiv.org/abs/math/0608183

U2 - 10.1353/ajm.0.0027

DO - 10.1353/ajm.0.0027

M3 - Article

SN - 0002-9327

VL - 130

SP - 1509

EP - 1534

JO - American Journal of Mathematics

JF - American Journal of Mathematics

IS - 6

ER -