The virtual K-theory of Quot schemes of surfaces

Abstract

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 smooth projective surfaces and dimension 1 quotients for surfaces X with p_g>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 three cases: Quot schemes of dimension 0 quotients, Hilbert schemes of points and curves over surfaces with p_g>0, 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.

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Cited by in corpus (7)

• Quot schemes of curves and surfaces: virtual classes, integrals, Euler characteristics
• Hyperquot schemes on curves: virtual class and motivic invariants
• Cosection localization and the Quot scheme ${\mathrm{Quot}}_S^l(\mathcal{E})$
• Wall-crossing for zero-dimensional sheaves and Hilbert schemes of points on Calabi-Yau 4-folds
• K-Theoretic Descendent Series for Hilbert Schemes of Points on Surfaces
• Wall-crossing for punctual Quot-schemes
• Equivariant Segre and Verlinde invariants for Quot schemes