Projective toric varieties as fine moduli spaces of quiver representations
Projective toric varieties as fine moduli spaces of quiver representations
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作为箭袋表示的精细模空间的射影复曲面簇
DOI：
10.1353/ajm.0.0027
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发表时间：
2006
影响因子：
1.7
通讯作者：
Gregory G. Smith
中科院分区：
文献类型：
--
作者：
Alastair Craw;Gregory G. Smith
本文证明了每个射影环面簇都是适当边界箭袋稳定表示的精细模空间。为了实现这一点，我们研究了箭袋 $Q$，其关系 $R$ 对应于有限维代数 $\mathop{\rm End}\nolimits( \textstyle\bigoplus\nolimits_{i=0}^{r} L_i )$ where ${\cal L} := ({\scr O}_X,L_1, \ldots, L_r)$ 是射影复曲面簇 $X$ 上的线束列表。箭袋 $Q$ 定义了一个平滑的射影复曲面簇，称为多线性级数 $|{\cal L}|$ 和映射 $X \longrightarrow |{\cal L}|$。我们为诱导图成为封闭嵌入提供了充分必要条件。因此，我们获得了射影复曲面簇的新几何商构造。在对 ${\cal L}$ 稍强的假设下，闭合嵌入将 $X$ 与边界箭袋 $(Q,R)$ 的稳定表示的精细模空间一起识别。
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 $(Q,R)$.