Volume 15, issue 1 (2011)

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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 $\delta$–nodal curves in a general $\delta$–dimensional linear system is given by a universal polynomial of degree $\delta$ in the four numbers ${L}^{2},\phantom{\rule{0.3em}{0ex}}L.{K}_{S},\phantom{\rule{0.3em}{0ex}}{K}_{S}^{2}$ and ${c}_{2}\left(S\right)$.

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 $\left(5\delta -1\right)$–very ample to $\delta$–very ample.

Keywords
Göttsche conjecture, Goettsche conjecture, counting nodal curves on surfaces
Mathematical Subject Classification 2000
Primary: 14C05, 14N10
Secondary: 14C20, 14N35
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