Segre classes of tautological bundles on Hilbert schemes of surfaces

TL;DR

This paper provides a new geometric proof for the top Segre classes of tautological bundles on Hilbert schemes of K3 surfaces and explores vanishing results for blow-ups, advancing understanding of these classes.

Contribution

It offers an alternative geometric proof for known results and extends the analysis to blow-ups of K3 surfaces, establishing new vanishing theorems.

Findings

New geometric proof of top Segre class results

Vanishing theorems for Segre classes on blow-ups of K3 surfaces

Determination of all top Segre classes for pairs (Σ, H)

Abstract

We first give an alternative proof, based on a simple geometric argument, of a result of Marian, Oprea and Pandharipande on top Segre classes of the tautological bundles on Hilbert schemes of $K3$ surfaces equipped with a line bundle. We then turn to the blow-up of $K3$ surface at one point and establish vanishing results for the corresponding top Segre classes in a certain range. This determines, at least theoretically, all top Segre classes of tautological bundles for any pair $(\Sigma,H),\,H\in {\rm Pic}\,\Sigma$.