Stable maps, multiplicities, and compactified Jacobians

TL;DR

This paper explores the relationships among deformation spaces, stable maps, and compactified Jacobians of singular curves, correcting previous claims about multiplicities and providing conditions for their validity.

Contribution

It corrects a prior statement on multiplicities in deformation spaces and establishes precise conditions for the behavior of these geometric objects.

Findings

Corrects a statement by Fantechi–Göttsche–van Straten on multiplicity

Provides a necessary and sufficient condition for the original claim

Analyzes numerical relations among deformation space, stable maps, and Jacobians

Abstract

Let $C$ be a complex projective integral curve with planar singularities. In this note, we study numerical relations among its versal deformation space, moduli space of stable maps, and compactified Jacobian. In particular, we correct a statement by Fantechi--G\"ottsche--van Straten on the multiplicity of the $\delta$-constant stratum of the versal deformation space at $[C]$. We also give a necessary and sufficient condition for the original claim to hold.