The degree-2 Abel--Jacobi map for nodal curves - I

TL;DR

This paper provides a modular description of the degree-2 Abel--Jacobi map for nodal curves, extending the generic fiber map to a compactified Jacobian in a smoothing family.

Contribution

It introduces a new modular framework for the Abel--Néron map in the context of nodal curve smoothings, extending previous degree-2 Abel--Jacobi maps.

Findings

Modular description of the Abel--Néron map in the context of nodal curves.

Extension of the degree-2 Abel--Jacobi map to a compactified Jacobian.

Application to smoothings of nodal curves and their Jacobians.

Abstract

Let $f\col\C\ra B$ be a regular local smoothing of a nodal curve. In this paper, we find a modular description of the Abel--N\'eron map having values in Esteves's fine compactified Jacobian and extending the degree-2 Abel--Jacobi map of the generic fiber of $f$