Higher rank Segre integrals over the Hilbert scheme of points

TL;DR

This paper extends the understanding of Segre classes of tautological bundles over Hilbert schemes of points on surfaces, providing explicit formulas for all ranks on K-trivial surfaces and for rank 2 on all surfaces, along with conjectures on Verlinde series.

Contribution

It generalizes Lehn's formula to all ranks on K-trivial surfaces and fully determines rank 2 Segre integrals on all surfaces, introducing new explicit formulas and conjectures.

Findings

Segre classes for all ranks on K-trivial surfaces are computed.

Full formulas for rank 2 Segre integrals over all surfaces are established.

Conjectural formulas for Verlinde Euler characteristics are proposed.

Abstract

Let S be a nonsingular projective surface. Each vector bundle V on S of rank s induces a tautological vector bundle over the Hilbert scheme of n points of S. When s=1, the top Segre classes of the tautological bundles are given by a recently proven formula conjectured in 1999 by M. Lehn. We calculate here the Segre classes of tautological bundles for all ranks s over all K-trivial surfaces. Furthermore, in rank s=2, the Segre integrals are determined for all surfaces, thus establishing a full analogue of Lehn's formula. We also give conjectural formulas for certain series of Verlinde Euler characteristics over the Hilbert schemes of points.