The virtual K-theory of Quot schemes of surfaces

TL;DR

This paper investigates the virtual K-theoretic invariants of Quot schemes on surfaces, conjectures their generating series are rational functions, and proves this in several cases, revealing new symmetries and relations with cobordism classes.

Contribution

It introduces conjectures on the rationality of generating series of virtual K-theoretic invariants and proves them for specific geometries, also discovering new symmetries in the invariants.

Findings

Rationality of generating series for punctual quotients on all smooth projective surfaces

Matching of Segre and Verlinde series in specific cases like Hilbert schemes and elliptic surfaces

New symmetry exchanging rank r and N in punctual Quot schemes

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.