## Geometry & Topology

### A short proof of the Göttsche conjecture

#### 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 $(5δ−1)$–very ample to $δ$–very ample.

#### Article information

Source
Geom. Topol., Volume 15, Number 1 (2011), 397-406.

Dates
Accepted: 12 December 2010
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.gt/1513732280

Digital Object Identifier
doi:10.2140/gt.2011.15.397

Mathematical Reviews number (MathSciNet)
MR2776848

Zentralblatt MATH identifier
1210.14011

#### Citation

Kool, Martijn; Shende, Vivek; Thomas, Richard P. A short proof of the Göttsche conjecture. Geom. Topol. 15 (2011), no. 1, 397--406. doi:10.2140/gt.2011.15.397. https://projecteuclid.org/euclid.gt/1513732280

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