Extending the Double Ramification Cycle using Jacobians

TL;DR

This paper proves that different approaches to extending the double ramification cycle in algebraic geometry are equivalent, unifying multiple constructions using Jacobians, Brill-Noether loci, and virtual fundamental classes.

Contribution

It establishes the equivalence of various methods for extending the double ramification cycle, connecting stack modifications, Jacobian loci, and rubber map techniques.

Findings

Different extension methods coincide in the untwisted case.

Unification of approaches simplifies understanding of double ramification cycles.

Results apply to compactified universal Jacobians and stable curves.

Abstract

We prove that the extension of the double ramification cycle defined by the first-named author (using modifications of the stack of stable curves) coincides with that defined by the last-two named authors (using an extended Brill-Noether locus on suitable compactified universal Jacobians). In particular, in the untwisted case we deduce that both of these extensions coincide with that constructed by Li and Graber-Vakil using a virtual fundamental class on a space of rubber maps.