Intersection numbers on the relative Hilbert schemes of points on surfaces

TL;DR

This paper derives formulas for intersection numbers on relative Hilbert schemes of points on surfaces, relating them to non-relative cases and extending known results for Euler classes of tangent bundles.

Contribution

It introduces a formula connecting relative and non-relative intersection numbers on Hilbert schemes, generalizing Carlsson-Okounkov's explicit Euler class formula.

Findings

Derived a formula relating relative and non-relative intersection numbers.

Extended Carlsson-Okounkov's explicit formula to the relative setting.

Provided new tools for studying intersection theory on Hilbert schemes.

Abstract

We study certain top intersection products on the Hilbert scheme of points on a nonsingular surface relative to an effective smooth divisor. We find a formula relating these numbers to the corresponding intersection numbers on the non-relative Hilbert schemes. In particular, we obtain a relative version of the explicit formula found by Carlsson-Okounkov for the Euler class of the twisted tangent bundle of the Hilbert schemes.