Euler number of the compactified Jacobian and multiplicity of rational curves

Details Euler number of the compactified Jacobian and multiplicity of rational curves Euler number of the compactified Jacobian and multiplicity of rational curves

1997-08-14

arxiv.org/abs/alg-geom/9708012v1

Mathematics

Algebraic Geometry

LaTeX, 16 pages with 1 figure

Scientific paper

We show that the Euler number of the compactified Jacobian of a rational curve $C$ with locally planar singularities is equal to the multiplicity of the $\delta$-constant stratum in the base of a semi-universal deformation of $C$. In particular, the multiplicity assigned by Yau, Zaslow and Beauville to a rational curve on a K3 surface $S$ coincides with the multiplicity of the normalisation map in the moduli space of stable maps to $S$.

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