Stable maps, multiplicities, and compactified Jacobians

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–Göttsche–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.