Universal Series for Hilbert Schemes and Strange Duality

TL;DR

This paper introduces universal power series related to Hilbert schemes of points on surfaces, connecting tautological sheaves and Euler characteristics, supported by computational verification and combinatorial proofs.

Contribution

It proposes conjectures linking two sets of universal power series on Hilbert schemes, extending to all smooth complex projective surfaces, with computational and combinatorial validation.

Findings

Verified conjectures for low order cases

Established a combinatorial proof involving Catalan numbers

Connected top Chern classes to universal power series

Abstract

We show how the "finite Quot scheme method" applied to Le Potier's strange duality on del Pezzo surfaces leads to conjectures (valid for all smooth complex projective surfaces) relating two sets of universal power series on Hilbert schemes of points on surfaces: those for top chern classes of tautological sheaves, and those for Euler characteristics of line bundles. We have verified these predictions computationally for low order. We then give an analysis of these conjectures in small ranks. We also give a combinatorial proof of a formula predicted by our conjectures: the top chern class of the tautological sheaf on $S^{[n]}$ associated to the structure sheaf of a point is equal to $(-1)^n$ times the $n$th Catalan number.