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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

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.

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