Universal polynomials for tautological integrals on Hilbert schemes

Rennemo, Jørgen

doi:10.2140/gt.2017.21.253

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Universal polynomials for tautological integrals on Hilbert schemes

Jørgen Vold Rennemo

Geometry & Topology 21 (2017) 253–314

DOI: 10.2140/gt.2017.21.253

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 $(X, L)$ .

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.

Keywords

Hilbert schemes, tautological bundles, Göttsche conjecture, counting singular divisors

Mathematical Subject Classification 2010

Primary: 14C05, 14N10, 14N35

References

Publication

Received: 25 November 2014

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

Volume 21, issue 1 (2017)