Multiplicativity of the double ramification cycle

TL;DR

This paper extends the double ramification cycle to the small b-Chow ring to restore its multiplicative relation over the entire moduli space of stable curves, providing evidence for a conjectured cycle equality.

Contribution

It introduces an extension of the double ramification cycle to the small b-Chow ring, restoring multiplicativity over all stable curves.

Findings

Restores multiplicative relation in the small b-Chow ring.

Provides evidence for the conjectured equality with a previously constructed cycle.

Extends the understanding of double ramification cycles beyond compact-type curves.

Abstract

The double ramification cycle satisfies a basic multiplicative relation DRC(a).DRC(b) = DRC(a).DRC(a + b) over the locus of compact-type curves, but this relation fails in the Chow ring of the moduli space of stable curves. We restore this relation over the moduli space of stable curves by introducing an extension of the double ramification cycle to the small b-Chow ring (the colimit of the Chow rings of all smooth blowups of the moduli space). We use this to give evidence for the conjectured equality between the (twisted) double ramification cycle and a cycle constructed in previous work by the second author.