Resumen de Projective toric varieties as fine moduli spaces of quiver representations

Alastair Craw, Gregory G. Smith

• 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 $(Q,R)$.