Verlinde series for Hirzebruch surfaces

TL;DR

This paper provides explicit formulas for Euler characteristics of line bundles on Hilbert schemes of points on surfaces, extending known results to Hirzebruch surfaces and general smooth projective surfaces, with enumerative applications.

Contribution

It introduces a new explicit formula for Euler characteristics on Hilbert schemes of points on Hirzebruch surfaces, generalizing to all smooth projective surfaces.

Findings

Explicit Euler characteristic formulas for line bundles on Hilbert schemes.

Enumeration of global sections for ample line bundles on Hirzebruch surfaces.

Extension of polytope-line bundle correspondence to Hirzebruch surfaces.

Abstract

We give an explicit formula for Euler characteristics of line bundles on the Hilbert scheme of points on $\mathbb{P}^1\times\mathbb{P}^1$. Combined with structural results of Ellingsrud, G\"ottsche, and Lehn, this determines the Euler characteristic of any line bundle on the Hilbert scheme of points on any smooth, projective surface. We also give an enumerative description of the dimensions of spaces of global sections of ample line bundles on Hilbert schemes of points on Hirzebruch surfaces, extending the polytope-line bundle correspondence on the underlying toric surface.