Euler number of the compactified Jacobian and multiplicity of rational curves
Abstract
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
.