Geometry & Topology

A short proof of the Göttsche conjecture

Martijn Kool, Vivek Shende, and Richard P Thomas

Full-text: Open access

Abstract

We prove that for a sufficiently ample line bundle L on a surface S, the number of δ–nodal curves in a general δ–dimensional linear system is given by a universal polynomial of degree δ in the four numbers L2,L.KS,KS2 and c2(S).

The technique is a study of Hilbert schemes of points on curves on a surface, using the BPS calculus of Pandharipande and the third author [J. Amer. Math. Soc. 23 (2010) 267–297] and the computation of tautological integrals on Hilbert schemes by Ellingsrud, Göttsche and Lehn [J. Algebraic Geom. 10 (2001) 81–100].

We are also able to weaken the ampleness required, from Göttsche’s (5δ1)–very ample to δ–very ample.

Article information

Source
Geom. Topol., Volume 15, Number 1 (2011), 397-406.

Dates
Received: 2 November 2010
Accepted: 12 December 2010
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513732280

Digital Object Identifier
doi:10.2140/gt.2011.15.397

Mathematical Reviews number (MathSciNet)
MR2776848

Zentralblatt MATH identifier
1210.14011

Subjects
Primary: 14C05: Parametrization (Chow and Hilbert schemes) 14N10: Enumerative problems (combinatorial problems)
Secondary: 14C20: Divisors, linear systems, invertible sheaves 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants [See also 53D45]

Keywords
Göttsche conjecture Goettsche conjecture counting nodal curves on surfaces

Citation

Kool, Martijn; Shende, Vivek; Thomas, Richard P. A short proof of the Göttsche conjecture. Geom. Topol. 15 (2011), no. 1, 397--406. doi:10.2140/gt.2011.15.397. https://projecteuclid.org/euclid.gt/1513732280


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