Compactifying the image of the Abel map

TL;DR

This paper investigates the closure of the Abel map's image within a compactified Jacobian for various types of singular curves, enhancing understanding of their geometric properties.

Contribution

It provides a detailed description of the closure of the Abel map's image for different classes of singular curves within the compactified Jacobian.

Findings

Closure of Abel map's image characterized for irreducible curves

Closure described for curves of compact type

Closure analyzed for binary curves

Abstract

Let $\alpha_X^{\underline d}$ be the Abel map of multidegree $\underline d$ of a singular curve $X$ of genus $g$. We describe the closure of ${\rm Im}\alpha_X^{\underline d}$ inside Caporaso's compactified Jacobian $\bar{P_X^d}$ for irreducible curves, curves of compact type and binary curves.