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 isolated singularity types and in all dimensions. So if $L$ is a sufficiently ample line bundle on a smooth variety $X$, in a general subsystem ${\mathbb{P}}^d\subset \left|L\right|$ of appropriate dimension the number of hypersurfaces with given isolated singularity types is a polynomial in the Chern numbers of $\left(X,L\right)$.

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.