Abel Maps for nodal curves via tropical geometry

TL;DR

This paper uses tropical and toric geometry to explicitly resolve Abel maps for nodal curves, providing a combinatorial approach to understand their structure and constructing all degree one Abel maps.

Contribution

It introduces a combinatorial method via tropical geometry to explicitly resolve Abel maps for nodal curves, including all degree one cases.

Findings

Explicit resolutions of Abel maps are obtained through combinatorial methods.

The approach applies to regular smoothings of nodal curves.

All degree one Abel maps are constructed and analyzed.

Abstract

We consider Abel maps for regular smoothing of nodal curves with values in the Esteves compactified Jacobian. In general, these maps are just rational, and an interesting question is to find an explicit resolution. We translate this problem into an explicit combinatorial problem by means of tropical and toric geometry. We show that the solution of the combinatorial problem gives rise to an explicit resolution of the Abel map. We are able to use this technique to construct and study all the Abel maps of degree one.