Euler number of the compactified Jacobian and multiplicity of rational curves

TL;DR

This paper establishes a precise relationship between the Euler number of the compactified Jacobian of a rational curve with singularities and the multiplicity of a specific deformation stratum, linking geometric invariants to deformation theory.

Contribution

It proves that the Euler number of the compactified Jacobian equals the multiplicity of the $ ext{delta}$-constant stratum, connecting it to known multiplicities in K3 surface enumerations.

Findings

Euler number equals the multiplicity of the $ ext{delta}$-constant stratum.

The multiplicity matches that assigned to rational curves on K3 surfaces.

Provides a geometric interpretation of deformation multiplicities.

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 $\delta$-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 $S$.