Holmes, David; Pixton, Aaron; Schmitt, Johannes

Multiplicativity of the Double Ramification Cycle

Doc. Math. 24, 545-562 (2019)
DOI: 10.25537/dm.2019v24.545-562
Communicated by Gavril Farkas

Summary

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 [\textit{F. Janda} et al., Publ. Math., Inst. Hautes Étud. Sci. 125, 221--266 (2017; Zbl 1370.14029)].

Mathematics Subject Classification

14H10, 14C15, 30F30

Keywords/Phrases

moduli space of curves, strata of differentials, double ramification cycles, tautological classes

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Affiliation

Holmes, David
Mathematisch Instituut, Universiteit Leiden, Postbus 9512 2300 RA, Leiden, Netherlands
Pixton, Aaron
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Schmitt, Johannes
Department of Mathematics, ETH Z\``urich, Raemistrasse 101, 8092 Z\''urich, Switzerland

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