Multiplicativity of the Double Ramification Cycle

Abstract

The double ramification cycle satisfies a basic multiplicative relation ${\mathrm{DRC}}_a\cdot {\mathrm{DRC}}_b={\mathrm{DRC}}_a\cdot {\mathrm{DRC}}_{a+b}$ 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 $\mathrm{b}$-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 $P_g^{d,k}(A)$ described by the second author in [F. Janda et al., Publ. Math., Inst. Hautes Étud. Sci. 125, 221–266 (2017; Zbl 1370.14029)].