Euler number of the compactified Jacobian and multiplicity of rational curves

Fantechi, Barbara; Göttsche, Lothar; van Straten, Duco

Euler number of the compactified Jacobian and multiplicity of rational curves

Authors

Fantechi, Barbara

Göttsche, Lothar

van Straten, Duco

Description

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 $δ$-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 $S$.
LaTeX, 16 pages with 1 figure