Journal of Geometry and Physics ( IF 1.5 ) Pub Date : 2021-02-06 , DOI: 10.1016/j.geomphys.2021.104154 Noah Arbesfeld , Drew Johnson , Woonam Lim , Dragos Oprea , Rahul Pandharipande
We study virtual invariants of Quot schemes parametrizing quotients of dimension at most 1 of the trivial sheaf of rank on nonsingular projective surfaces. We conjecture that the generating series of virtual -theoretic invariants are given by rational functions. We prove rationality for several geometries including punctual quotients for all surfaces and dimension 1 quotients for surfaces with . We also show that the generating series of virtual cobordism classes can be irrational.
Given a -theory class on of rank , we associate natural series of virtual Segre and Verlinde numbers. We show that the Segre and Verlinde series match in the following cases:
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[(i)] Quot schemes of dimension 0 quotients,
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[(ii)] Hilbert schemes of points and curves over surfaces with ,
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[(iii)] Quot schemes of minimal elliptic surfaces for quotients supported on fiber classes.
中文翻译:
虚拟的 曲面的报价方案理论
我们研究Quot方案的虚拟不变量,将参数的商分解为等级的平凡捆中最多1个 在非奇异的投影面上。我们推测虚拟的生成系列-理论不变量由有理函数给出。我们证明了几种几何的合理性,包括所有曲面的守时商和曲面的1维商 和 。我们还表明,虚拟的Cobordism类的生成系列可能是不合理的。
给定一个 -理论课 等级 ,我们将自然的虚拟Segre和Verlinde数列相关联。我们显示在以下情况下Segre和Verlinde系列匹配:
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[(i)] 0维商的报价方案,
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[(ii)]曲面上具有点和曲线的希尔伯特方案 ,
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[(iii)]光纤类别上所支持商的最小椭圆表面的报价方案。