Extending the double ramification cycle by resolving the Abel-Jacobi map

TL;DR

This paper develops a universal resolution of the Abel-Jacobi map to extend the double ramification cycle over the entire moduli space of stable curves, connecting it with existing constructions via virtual fundamental classes.

Contribution

It introduces a universal resolution of the Abel-Jacobi map, enabling the extension of the double ramification cycle to all stable curves, unifying different approaches.

Findings

The extended cycle matches Li, Graber, Vakil's construction in the non-twisted case.

The resolution is universal and works over the entire moduli space.

The approach bridges classical and modern techniques in moduli theory.

Abstract

Over the moduli space of smooth curves, the double ramification cycle can be defined by pulling back the unit section of the universal jacobian along the Abel-Jacobi map. This breaks down over the boundary since the Abel-Jacobi map fails to extend. We construct a `universal' resolution of the Abel-Jacobi map, and thereby extend the double ramification cycle to the whole of the moduli of stable curves. In the non-twisted case, we show that our extension coincides with the cycle constructed by Li, Graber, Vakil via a virtual fundamental class on a space of rubber maps.