Geometry & Topology

Universal polynomials for tautological integrals on Hilbert schemes

Jørgen Rennemo

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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 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.

Article information

Source
Geom. Topol., Volume 21, Number 1 (2017), 253-314.

Dates
Received: 25 November 2014
Revised: 15 December 2015
Accepted: 15 January 2016
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1510859134

Digital Object Identifier
doi:10.2140/gt.2017.21.253

Mathematical Reviews number (MathSciNet)
MR3608714

Zentralblatt MATH identifier
06687807

Subjects
Primary: 14C05: Parametrization (Chow and Hilbert schemes) 14N10: Enumerative problems (combinatorial problems) 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants [See also 53D45]

Keywords
Hilbert schemes tautological bundles Göttsche conjecture counting singular divisors

Citation

Rennemo, Jørgen. Universal polynomials for tautological integrals on Hilbert schemes. Geom. Topol. 21 (2017), no. 1, 253--314. doi:10.2140/gt.2017.21.253. https://projecteuclid.org/euclid.gt/1510859134


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