Universal Polynomials for Tautological Integrals on Hilbert Schemes

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 Göttsche 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.

44 pages, minor changes and corrections

References in corpus (5)

• Zero dimensional Donaldson-Thomas invariants of threefolds
• Euler characteristics of general linear sections and polynomial Chern classes
• A Proof of the Göttsche-Yau-Zaslow Formula
• Universal polynomials for singular curves on surfaces
• Characteristic classes of Hilbert schemes of points via symmetric products

Cited by in corpus (13)

• Discriminants in the Grothendieck Ring
• Tautological integrals on curvilinear Hilbert schemes
• Enumeration of curves with one singular point
• Sheaves on surfaces and virtual invariants
• Enumeration of singular varieties with tangency conditions
• Tautological Integrals on Hilbert Schemes of Points on Curves
• Remarks on the derived McKay correspondence for Hilbert schemes of points and tautological bundles
• Determinantal representations of the cubic discriminant
• Counting curves in a linear system with upto eight singular points
• The Kleiman-Piene Conjecture and node polynomials for plane curves in ${\mathbb{P}}^3$
• Universal polynomials for counts of secant planes to projective curves
• An inclusion-exclusion principle for tautological sheaves on Hilbert schemes of points
• Curve Counting à la Göttsche