#### 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