Projective toric varieties as fine moduli spaces of quiver representations

Craw, A. and Smith, G.G. (2008) Projective toric varieties as fine moduli spaces of quiver representations. American Journal of Mathematics, 130(6), pp. 1509-1534. (doi: 10.1353/ajm.0.0027)

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 $\mathop{\rm End}\nolimits( \textstyle\bigoplus\nolimits_{i=0}^{r} L_i )$ where ${\cal L} := ({\scr O}_X,L_1, \ldots, L_r)$ 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 $|{\cal L}|$, and a map $X \longrightarrow |{\cal L}|$. 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 ${\cal L}$, the closed embedding identifies $X$ with the fine moduli space of stable representations for the bound quiver $(Q,R)$.

• Item Type: Articles
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Craw, Dr Alastair
• Authors: Craw, A., and Smith, G.G.
• Subjects: Q Science > QA Mathematics
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: American Journal of Mathematics
• Journal Abbr.: Amer. J. Math.
• ISSN: 0002-9327
• ISSN (Online): 1080-6377