Geometry & Topology 15 (2011) 397–406

DOI: 10.2140/gt.2011.15.397

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 . K_S,K_S² 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 (5δ−1)–very ample to δ–very ample.

Keywords

Göttsche conjecture, Goettsche conjecture, counting nodal curves on surfaces

Mathematical Subject Classification

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