Projective toric varieties as fine moduli spaces of quiver representations

Details Projective toric varieties as fine moduli spaces of quiver representations Projective toric varieties as fine moduli spaces of quiver representations

2006-08-07

arxiv.org/abs/math/0608183v2

American Journal of Mathematics 130 (2008) 1509-1534

Mathematics

Algebraic Geometry

revised version: improved exposition, corrected typos and other minor changes

Scientific paper

10.1353/ajm.0.0027

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 $(Q,R)$.

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