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Universal polynomials for tautological integrals on Hilbert schemes

Jørgen Vold Rennemo

Geometry & Topology 21 (2017) 253–314

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 d |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.

Hilbert schemes, tautological bundles, Göttsche conjecture, counting singular divisors
Mathematical Subject Classification 2010
Primary: 14C05, 14N10, 14N35
Received: 25 November 2014
Revised: 15 December 2015
Accepted: 15 January 2016
Published: 10 February 2017
Proposed: Jim Bryan
Seconded: Lothar Göttsche, Ronald Stern
Jørgen Vold Rennemo
All Souls College
United Kingdom