Multiplicativity of the Double Ramification Cycle

  • David Holmes

    Mathematisch Instituut, Universiteit Leiden, Postbus 9512 2300 RA, Leiden, Netherlands
  • Aaron Pixton

    Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
  • Johannes Schmitt

    Department of Mathematics, ETH Zürich, Raemistrasse 101, 8092 Zürich, Switzerland
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The double ramification cycle satisfies a basic multiplicative relation DRCaDRCb=DRCaDRCa+b\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 b\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 Pgd,k(A)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)].

Cite this article

David Holmes, Aaron Pixton, Johannes Schmitt, Multiplicativity of the Double Ramification Cycle. Doc. Math. 24 (2019), pp. 545–562

DOI 10.4171/DM/688