Higher rank Segre integrals over the Hilbert scheme of points

Alina Marian,Dragos Oprea,Rahul Pandharipande

doi:10.4171/jems/1149

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.

Highlights

Due to the rank shift (6), the Serre duality symmetry s → −s on the Verlinde side translates into a conjectural transformation rule for the remaining unknown Segre universal series A3, A4, as s → −s − 2
The evaluation of A0, A1, and A2 for Chern classes follows by regarding the Segre integrals of Theorem 1 as functions on the K-theory of the surface S which depend polynomially on rank α = s, c1(α)2, c2(α), see [EGL]
Afterwards we show Theorem 2: the integrals calculated in Theorem 4 give the rank 2 series for all surfaces

Summary

Introduction

The generating series of higher rank Segre integrals was determined in [MOP2]. Finding the general expression for the unknown series B3 and B4 for arbitrary r, determining all rank 1 Verlinde numbers, is a central question in the enumerative theory of Hilbert schemes of points on surfaces. Due to the rank shift (6), the Serre duality symmetry s → −s on the Verlinde side translates into a conjectural transformation rule for the remaining unknown Segre universal series A3, A4, as s → −s − 2. In the case of the moduli of K3 surfaces, a strategy for κ-relations was laid out in [MOP1] via the study of the virtual class of the Quot scheme; a different approach via Gromov-Witten theory was pursued in [PY].

K3 surfaces

Abelian and Enriques surfaces

K3 blowups