On the moduli of reflexive sheaves on a surface with rational double points
On the moduli of reflexive sheaves on a surface with rational double points
复制标题
有理双点面上自反滑轮的模量
DOI：
10.1007/bf01934318
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发表时间：
1992
影响因子：
1.4
通讯作者：
A. Ishii
中科院分区：
文献类型：
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作者：
A. Ishii
设X是一个有至多有理双点的投影曲面。我们将考虑x上的（稳定）自反轴的模空间。考虑这样的曲面的理由如下：首先，Simpson最近证明了我们甚至可以在奇异变异上构造稳定束的模格式[S]。然后我们要研究它们与非奇异情况的区别。有理双点似乎是最容易治疗的。另一个更为关键的原因是一般型曲面的规范模型就是这样的曲面。稳定轮轴的模量格式取决于变异上偏振的选择（如果变异的尺寸大于1）。因此，对于一般类型的曲面，很自然地要考虑它相对于规范束的稳定性。然而，两极分化应该是充分的，因此我们必须研究规范模型。我们考虑的问题如下。(1)在奇异点处X上具有给定类型自反模的稳定自反束的存在性。(2)模空间如何反映曲面的奇异性。问题(1)的结果在定理2.4中表示。一个有理双点上的自反模的同构类由它的秩和它的回拉到最小非规格化的第一个Chem类决定。定理2.4表述如下：
Let X be a projective surface with at most rational double points. We shall consider the moduli space of (stable) reflexive sheaves on X. The reasons to consider such surfaces are as follows. First, Simpson has recently shown that we can construct the moduli schemes of stable sheaves even on singular varieties [S]. Then we want to study how they differ from non-singular cases. Rational double points seem to be the easiest to treat. Another and more crucial reason is that the canonical models of surfaces of general type are such surfaces. The moduli scheme of stable sheaves depends on the choice of the polarization on the variety (if its dimension is greater than 1). Hence, for a surface of general type, it is natural to consider the stability with respect to the canonical sheaf. However, the polarization should be ample and hence we have to work on canonical models. The questions we consider are the following.(1) Existence of stable reflexive sheaves on X with given types of reflexive modules at singular points.(2) How the moduli space reflects the singularity of the surface. The result for the question (1) is stated in Theorem 2.4. The isomorphism class of a reflexive module over a rational double point is determined by its rank and the first Chem class of its pull-back to the minimal desingularization [AV]. Theorem 2.4 is stated as follows.