Euler number of the compactified Jacobian and multiplicity of rational curves

In this paper we show that the Euler number of the compactified Jacobian (J ) over bar C of a rational curve C with locally planar singularities is equ al to the multiplicity of the delta-constant stratum in the base of a semi- universal deformation of C. The number e((J) over bar C) is the multiplicit y assigned by Beauville to C in his proof of the formula, proposed by Yau a nd Zaslow, for the number of rational curves on a K3 surface X. We prove th at e((J) over bar C) also coincides with the multiplicity of the normalisat ion map of C in the moduli space of stable maps to X.