Mori dream spaces as fine moduli of quiver representations
Mori dream spaces as fine moduli of quiver representations
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DOI：
10.1016/j.jpaa.2012.06.014
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发表时间：
2011-04
影响因子：
0.8
通讯作者：
Alastair Craw;Dorothy Winn
中科院分区：
文献类型：
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作者：
Alastair Craw;Dorothy Winn
Mori Dream Spaces和他们的考克斯环自从十多年前由Hu-Keel [11]介绍以来一直是人们极大兴趣的主题。从几何方面来看，这些品种享有的财产，所有的业务的森计划可以进行的变化GIT商，而从代数方面，获得明确的介绍考克斯环本身是一个有趣的问题。Mori Dream Spaces的例子包括任意维数的Q-阶乘投射环面簇、球面簇和log Fano簇。本文利用箭图表示理论研究Mori Dream空间上的多重线性级数。我们的主要结果构造了Mori Dream空间作为有界箭图的稳定表示的细模空间，从而推广了Craw-Smith [6]关于投射复曲面簇的结果.设X是投射簇，L=（OX，L1，. Lr）X上不同的有效线丛的集合。相伴的多重线性级数|L|是一个光滑的投射环面品种，提供了一个多级模拟的经典线性系列的一个单一的线丛。构建|L|首先定义L的截集的界为一个有限的非循环的λ Q，以及路代数kQ中的一个关系理想JL，其中kQ/JL同构于End（λ 0≤ i≤ r Li）.先不考虑关系的理想，多级线性级数|L|被定义为作为Q的具有维数向量（1，...，1)对于特殊稳定性参数θ=（− r，1，...，1）。图中的路径产生于形式为H 0（X，Lj <$L− 1）的截面空间
Mori Dream Spaces and their Cox rings have been the subject of a great deal of interest since their introduction by Hu–Keel [11] over a decade ago. From the geometric side, these varieties enjoy the property that all operations of the Mori programme can be carried out by variation of GIT quotient, while from the algebraic side, obtaining an explicit presentation of the Cox ring is an interesting problem in itself. Examples of Mori Dream Spaces include Q-factorial projective toric varieties, spherical varieties and log Fano varieties of arbitrary dimension. In this paper we use the representation theory of quivers to study multigraded linear series on Mori Dream Spaces. Our main results construct Mori Dream Spaces as fine moduli spaces of ϑ-stable representations of bound quivers for a special stability parameter ϑ, thereby extending results of Craw–Smith [6] for projective toric varieties.Let X be a projective variety and L=(OX, L1,..., Lr) a collection of distinct, effective line bundles on X. The associated multigraded linear series| L| is a smooth projective toric variety that provides a multigraded analogue of the classical linear series of a single line bundle. To construct| L| one first defines the bound quiver of sections of L to be a finite, acyclic quiver Q together with an ideal of relations JL in the path algebra kQ for which kQ/JL is isomorphic to End ( 0≤ i≤ r Li). Setting aside the ideal of relations for now, the multigraded linear series| L| is defined to be the toric quiver variety obtained as the fine moduli space of ϑ-stable representations of Q with dimension vector (1,..., 1) for the special stability parameter ϑ=(− r, 1,..., 1). Paths in the quiver arise from spaces of sections of the form H0 (X, Lj⊗ L− 1