Tautological Integrals on Hilbert Schemes of Points on Curves

TL;DR

This paper formulates a conjecture on generating series of Chern numbers for tautological bundles on Hilbert schemes of points on curves, proving specific cases and computing explicit series similar to Lehn's conjecture for surfaces.

Contribution

It introduces a new conjecture for Hilbert schemes on curves and verifies special cases, providing explicit formulas for Segre class integrals.

Findings

Confirmed the rank 1 and -1 cases of the conjecture

Computed explicit generating series for Segre classes on curves

Established structural similarities with Lehn's conjecture for surfaces

Abstract

We propose a conjecture on the generating series of Chern numbers of tautological bundles on Hilbert schemes of points on curves and establish the rank 1 and rank -1 case of this conjecture. Thus we compute explicitly the generating series of integrals of the Segre classes of tautological bundles of line bundles on curves, which has a similar structure as Lehn's conjecture for surfaces.