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Publications mathématiques de l'IHÉS

, Volume 125, Issue 1, pp 221–266 | Cite as

Double ramification cycles on the moduli spaces of curves

  • F. Janda
  • R. Pandharipande
  • A. Pixton
  • D. Zvonkine
Article

Abstract

Curves of genus \(g\) which admit a map to \(\mathbf {P}^{1}\) with specified ramification profile \(\mu\) over \(0\in \mathbf {P}^{1}\) and \(\nu\) over \(\infty\in \mathbf {P}^{1}\) define a double ramification cycle \(\mathsf{DR}_{g}(\mu,\nu)\) on the moduli space of curves. The study of the restrictions of these cycles to the moduli of nonsingular curves is a classical topic. In 2003, Hain calculated the cycles for curves of compact type. We study here double ramification cycles on the moduli space of Deligne-Mumford stable curves.

The cycle \(\mathsf{DR}_{g}(\mu,\nu)\) for stable curves is defined via the virtual fundamental class of the moduli of stable maps to rubber. Our main result is the proof of an explicit formula for \(\mathsf{DR}_{g}(\mu,\nu)\) in the tautological ring conjectured by Pixton in 2014. The formula expresses the double ramification cycle as a sum over stable graphs (corresponding to strata classes) with summand equal to a product over markings and edges. The result answers a question of Eliashberg from 2001 and specializes to Hain’s formula in the compact type case.

When \(\mu=\nu=\emptyset\), the formula for double ramification cycles expresses the top Chern class \(\lambda_{g}\) of the Hodge bundle of \(\overline {\mathcal{M}}_{g}\) as a push-forward of tautological classes supported on the divisor of non-separating nodes. Applications to Hodge integral calculations are given.

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Copyright information

© IHES and Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  • F. Janda
    • 1
  • R. Pandharipande
    • 2
  • A. Pixton
    • 3
  • D. Zvonkine
    • 4
  1. 1.Department of MathematicsUniversity of MichiganAnn ArborUSA
  2. 2.Departement MathematikETH ZürichZürichSwitzerland
  3. 3.Department of MathematicsMITCambridgeUSA
  4. 4.Institut de Mathématiques de JussieuCNRSParisFrance

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