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The virtual K-theory of Quot schemes of surfaces
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 N on nonsingular projective surfaces. We conjecture that the generating series of virtual K-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 X with pg>0. We also show that the generating series of virtual cobordism classes can be irrational.

Given a K-theory class on X of rank r, we associate natural series of virtual Segre and Verlinde numbers. We show that the Segre and Verlinde series match in the following cases:

[(i)] Quot schemes of dimension 0 quotients,

[(ii)] Hilbert schemes of points and curves over surfaces with pg>0,

[(iii)] Quot schemes of minimal elliptic surfaces for quotients supported on fiber classes.

Moreover, for punctual quotients of the trivial sheaf of rank N, we prove a new symmetry of the Segre/Verlinde series exchanging r and N. The Segre/Verlinde statements have analogues for punctual Quot schemes over curves.



中文翻译:

虚拟的 ķ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 ñ在非奇异的投影面上。我们推测虚拟的生成系列ķ-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XpG>0。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 ķ-理论课 X 等级 [R,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:

[(i)] 0维商的报价方案,

[(ii)]曲面上具有点和曲线的希尔伯特方案 pG>0

[(iii)]光纤类别上所支持商的最小椭圆表面的报价方案。

而且,对于平凡的等级的守时商 ñ,我们证明了Segre / Verlinde级数交换的新对称性 [Rñ。Segre / Verlinde语句具有曲线上的准时报价方案的类似物。

更新日期:2021-02-19
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