A short proof of the Göttsche conjecture

Details A short proof of the Göttsche conjecture A short proof of the Göttsche conjecture

2010-10-15

arxiv.org/abs/1010.3211v2

Geom.Topol.15:397-406,2011

Mathematics

Algebraic Geometry

Minor corrections. Submitted version. 8 pages

Scientific paper

10.2140/gt.2011.15.397

We prove that for a sufficiently ample line bundle L on a surface S, the number of d-nodal curves in a general d-dimensional linear system is given by a universal polynomial of degree d 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 \cite{PT3} and the computation of tautological integrals on Hilbert schemes by Ellingsrud, G\"ottsche and Lehn. We are also able to weaken the ampleness required, from G\"ottsche's (5d-1)-very ample to d-very ample.

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