Projective toric varieties as fine moduli spaces of quiver representations

Alastair Craw, Gregory G. Smith

Research output: Contribution to journalArticle

18 Citations (Scopus)

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 $\bigl(\bigoplus_{i=0}^{r} L_i \bigr)$ 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 $(Q,R)$.
Original languageEnglish
Pages (from-to)1509-1534
JournalAmerican Journal of Mathematics
Volume130
Issue number6
DOIs
Publication statusPublished - Dec 2008

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