Universal Polynomials for Tautological Integrals on Hilbert Schemes

TL;DR

This paper demonstrates that tautological integrals on Hilbert schemes can be expressed as universal polynomials in Chern numbers across all dimensions, enabling broad applications in enumerative geometry and confirming a generalized G"ottsche Conjecture.

Contribution

It introduces universal polynomials for tautological integrals on Hilbert schemes, extending results to all dimensions and arbitrary geometric subsets, and proves a generalized G"ottsche Conjecture.

Findings

Universal polynomials express tautological integrals in all dimensions.

Validation of a generalized G"ottsche Conjecture for all singularity types.

Polynomial formulas for counting hypersurfaces with specified singularities.

Abstract

We show that tautological integrals on Hilbert schemes of points can be written in terms of universal polynomials in Chern numbers. The results hold in all dimensions, though they strengthen known results even for surfaces by allowing integrals over arbitrary "geometric" subsets (and their Chern-Schwartz-MacPherson classes). We apply this to enumerative questions, proving a generalised G\"ottsche Conjecture for all singularity types and in all dimensions. So if L is a sufficiently ample line bundle on a smooth variety X, in a general subsystem P^d of |L| of appropriate dimension the number of hypersurfaces with given singularity types is a polynomial in the Chern numbers of (X,L). When X is a surface, we get similar results for the locus of curves with fixed "BPS spectrum" in the sense of stable pairs theory.