Volume 15, issue 1 (2011)

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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
A short proof of the Göttsche conjecture

Martijn Kool, Vivek Shende and Richard P Thomas

Geometry & Topology 15 (2011) 397–406
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.

Keywords
Göttsche conjecture, Goettsche conjecture, counting nodal curves on surfaces
Mathematical Subject Classification 2000
Primary: 14C05, 14N10
Secondary: 14C20, 14N35
References
Publication
Received: 2 November 2010
Accepted: 12 December 2010
Published: 1 March 2011
Proposed: Lothar Göttsche
Seconded: Jim Bryan, Gang Tian
Authors
Martijn Kool
Department of Mathematics
Imperial College
180 Queens Gate
London SW7 2AZ
UK
Vivek Shende
Department of Mathematics
Princeton University
Princeton NJ 08540
USA
Richard P Thomas
Department of Mathematics
Imperial College
180 Queens Gate
London SW7 2AZ
UK