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


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


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


  • 1. Alessandro Chiodo. Néron models of Pic$^0$ via Pic$^0$, 2015. arxiv 1509.06483.
  • 2. Emily Clader and Felix Janda. Pixton's double ramification cycle relations. Geom. Topol., 22(2):1069-1108, 2018. DOI 10.2140/gt.2018.22.1069; zbl 1387.14079; MR3748684; arxiv 1601.02871.
  • 3. Tom Graber and Rahul Pandharipande. Constructions of nontautological classes on moduli spaces of curves. Michigan Math. J., 51(1):93-109, 2003. DOI 10.1307/mmj/1049832895; zbl 1079.14511; MR1960923; arxiv math/0104057.
  • 4. Tom Graber and Ravi Vakil. Relative virtual localization and vanishing of tautological classes on moduli spaces of curves. Duke Math. J., 130(1):1-37, 2005. DOI 10.1215/S0012-7094-05-13011-3; zbl 1088.14007; MR2176546; arxiv math.AG/0309227.
  • 5. Samuel Grushevsky and Dmitry Zakharov. The double ramification cycle and the theta divisor. Proc. Amer. Math. Soc., 142(no. 12):4053-4064, 2014. DOI 10.1090/S0002-9939-2014-12153-8; zbl 1327.14132; MR3266977; arxiv 1206.7001.
  • 6. Samuel Grushevsky and Dmitry Zakharov. The zero section of the universal semiabelian variety, and the double ramification cycle. Duke Math J., 163(5):889-1070, 2014. DOI 10.1215/00127094-26444575; zbl 1302.14039; MR3189435; arxiv 1206.3534.
  • 7. Richard Hain. Normal functions and the geometry of moduli spaces of curves. In G. Farkas and I. Morrison, editors, Handbook of Moduli, Volume I. Advanced Lectures in Mathematics, Volume XXIV. International Press, Boston, 2013. zbl 1322.14049; MR3184171; arxiv 1102.4031.
  • 8. David Holmes, Jesse Leo Kass, and Nicola Pagani. Extending the double ramification cycle using Jacobians. Eur. J. Math., 4(3):1087-1099, 2018. DOI 10.1007/s40879-018-0256-7; MR3851130; arxiv 1712.07098.
  • 9. David Holmes. A Néron model of the universal jacobian, 2014. arxiv 1412.2243.
  • 10. David Holmes. Extending the double ramification cycle by resolving the Abel-Jacobi map. Preprint, 2017. arxiv 1707.02261.
  • 11. David Holmes. Néron models of jacobians over base schemes of dimension greater than 1. J. Reine Angew. Math., 747:109-145, 2019. DOI 10.1515/crelle-2016-0014; zbl 07012669; MR3905131.
  • 12. Eleny-Nicoleta Ionel. Topological recursive relations in $H^{2g}(M_{g,n})$. Invent. Math., 148(3):627-658, 2002. DOI 10.1007/s002220100205; zbl 1056.14076; MR1908062; arxiv math/9908060.
  • 13. Luc Illusie and Michael Temkin. Exposé VIII. Gabber's modification theorem (absolute case). Astérisque, (363-364):103-160, 2014. Travaux de Gabber sur l'uniformisation locale et la cohomologie étale des schémas quasi-excellents. zbl 1327.14071; MR3329777.
  • 14. Felix Janda, Rahul Pandharipande, Aaron Pixton, and Dimitri Zvonkine. Double ramification cycles on the moduli spaces of curves. Publications mathématiques de l'IHÉS, 125(1):221-266, 2017. DOI 10.1007/s10240-017-0088-x; zbl 1370.14029; MR3668650; arxiv 1602.04705.
  • 15. Kazuya Kato. Logarithmic degeneration and Dieudonné theory. Preprint, 1989.
  • 16. Fumiharu Kato. Log smooth deformation theory. Tohoku Mathematical Journal, Second Series, 48(3):317-354, 1996. DOI 10.2748/tmj/1178225336; zbl 0876.14007; MR1404507; arxiv alg-geom/9406004.
  • 17. Jesse Leo Kass and Nicola Pagani. The stability space of compactified universal Jacobians. To appear in Trans. Amer. Math. Soc., 2017. arxiv 1707.02284.
  • 18. Jun Li. Stable morphisms to singular schemes and relative stable morphisms. J. Differential Geom., 57(3):509-578, 2001. DOI 10.4310/jdg/1090348132; zbl 1076.14540; MR1882667.
  • 19. Jun Li. A degeneration formula of GW-invariants. J. Differential Geom., 60(2):199-293, 2002. DOI 10.4310/jdg/1090351102; zbl 1063.14069; MR1938113; arxiv math/0110113.
  • 20. Martin Christian Olsson. Log algebraic stacks and moduli of log schemes. ProQuest LLC, Ann Arbor, MI, 2001. Thesis (Ph.D.), University of California, Berkeley. MR2702292.
  • 21. Vyacheslav Vladimirovich Shokurov. 3-fold log models. Journal of Mathematical Sciences, 81(3):2667-2699, 1996. DOI 10.1007/BF02362335; zbl 0873.14014; MR1420223.
  • 22. Vyacheslav Vladimirovich Shokurov. Prelimiting flips. Proc. Steklov Inst. Math., 240(0):82-219, 2003. zbl 1082.14019; MR1993750.
  • 23. Sarah Scherotzke, Nicolò Sibilla, and Mattia Talpo. On a logarithmic version of the derived McKay correspondence. Compos. Math., 154(12):2534-2585, 2018. DOI 10.1112/S0010437X18007431; zbl 07036912; MR3875460; arxiv 1612.08961.
  • 24. The Stacks Project Authors. Stacks project. http://stacks.math.columbia.edu, 2013.
  • 25. Michael Temkin. Functorial desingularization of quasi-excellent schemes in characteristic zero: the nonembedded case. Duke Math. J., 161(11):2207-2254, 2012. DOI 10.1215/00127094-1699539; zbl 1285.14015; MR2957701; arxiv 0904.1592.


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 Zürich, Raemistrasse 101, 8092 Zürich, Switzerland