### 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 language | English |
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Pages (from-to) | 1509-1534 |

Journal | American Journal of Mathematics |

Volume | 130 |

Issue number | 6 |

DOIs | |

Publication status | Published - Dec 2008 |

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## Cite this

Craw, A., & G. Smith, G. (2008). Projective toric varieties as fine moduli spaces of quiver representations.

*American Journal of Mathematics*,*130*(6), 1509-1534. https://doi.org/10.1353/ajm.0.0027