#### Volume 21, issue 1 (2017)

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Universal polynomials for tautological integrals on Hilbert schemes

### Jørgen Vold Rennemo

Geometry & Topology 21 (2017) 253–314
##### 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 ${ℙ}^{d}\subset |L|$ 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.

##### Keywords
Hilbert schemes, tautological bundles, Göttsche conjecture, counting singular divisors
##### Mathematical Subject Classification 2010
Primary: 14C05, 14N10, 14N35
##### Publication
Revised: 15 December 2015
Accepted: 15 January 2016
Published: 10 February 2017
Proposed: Jim Bryan
Seconded: Lothar Göttsche, Ronald Stern
##### Authors
 Jørgen Vold Rennemo All Souls College Oxford OX1 4AL United Kingdom