Logarithmic compactification of the Abel-Jacobi section

TL;DR

This paper introduces a logarithmic compactification of the Abel-Jacobi map on moduli spaces of curves, extending its domain and connecting it to double ramification cycles using tropical and logarithmic geometry.

Contribution

It constructs a new modular compactification of the Abel-Jacobi map that extends over the boundary of the moduli space of stable curves, incorporating tropical and logarithmic techniques.

Findings

Provides a modular description of the extended Abel-Jacobi map.

Recovers the double ramification cycle within the new framework.

Describes the deformation theory and virtual fundamental class of the modified space.

Abstract

Given a smooth curve with weighted marked points, the Abel-Jacboi map produces a line bundle on the curve. This map fails to extend to the full boundary of the moduli space of stable pointed curves. Using logarithmic and tropical geometry, we describe a modular modification of the moduli space of curves over which the Abel-Jacobi map extends. We also describe the attendant deformation theory and virtual fundamental class of this moduli space. This recovers the double ramification cycle, as well as variants associated to differentials.