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arXiv · 1010.3211

A short proof of the Göttsche conjecture

M. Kool·V. Shende·R. P. Thomas
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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 $L^2,\,L.K_S,\,K_S^2$ and $c_2(S)$. The technique is a study of Hilbert schemes of points on curves on a surface, using the BPS calculus of [PT3] and the computation of tautological integrals on Hilbert schemes by Ellingsrud, Göttsche and Lehn. We are also able to weaken the ampleness required, from Göttsche's $(5δ-1)$-very ample to $δ$-very ample.

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BibTeXRIS

M. Kool, V. Shende, R. P. Thomas. 2014-03-22. A short proof of the Göttsche conjecture. https://doi.org/10.2140/gt.2011.15.397

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