Degree-2 Abel maps and hyperelleptic curves

TL;DR

This paper resolves the degree-2 Abel map for nodal curves using tropical geometry and characterizes when it is non-injective, linking this property to hyperelliptic curves.

Contribution

It provides a resolution of the degree-2 Abel map for nodal curves and characterizes non-injectivity in terms of hyperelliptic properties.

Findings

Resolved the degree-2 Abel map for nodal curves.

Characterized when the Abel map is not injective.

Linked non-injectivity to hyperelliptic curves.

Abstract

In this paper we resolve the degree-2 Abel map for nodal curves. Our results are based on a previous work of the authors reducing the problem of the resolution of the Abel map to a combinatorial problem via tropical geometry. As an application, we characterize when the (symmetrized) degree-2 Abel map is not injective, a property that, for a smooth curve, is equivalent to the curve being hyperelliptic.