Double ramification cycles on the moduli spaces of curves

Janda, F.; Pandharipande, R.; Pixton, A.; Zvonkine, D.

doi:10.1007/s10240-017-0088-x

Janda, F.¹ ; Pandharipande, R.² ; Pixton, A.³ ; Zvonkine, D.⁴

¹ Department of Mathematics, University of Michigan Ann Arbor USA
² Departement Mathematik, ETH Zürich Zürich Switzerland
³ Department of Mathematics, MIT Cambridge USA
⁴ Institut de Mathématiques de Jussieu, CNRS Paris France

Publications Mathématiques de l'IHÉS, Tome 125 (2017), pp. 221-266.

Résumé

Curves of genus $g$ which admit a map to $P^{1}$ with specified ramification profile $μ$ over $0 \in P^{1}$ and $ν$ over $\infty \in P^{1}$ define a double ramification cycle ${DR}_{g} (μ, ν)$ 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 ${DR}_{g} (μ, ν)$ 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 ${DR}_{g} (μ, ν)$ 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 $μ = ν = \emptyset$ , the formula for double ramification cycles expresses the top Chern class $λ_{g}$ of the Hodge bundle of ${\overline{M}}_{g}$ as a push-forward of tautological classes supported on the divisor of non-separating nodes. Applications to Hodge integral calculations are given.

MR Zbl

DOI : 10.1007/s10240-017-0088-x

Affiliations des auteurs :

Janda, F. ¹ ; Pandharipande, R. ² ; Pixton, A. ³ ; Zvonkine, D. ⁴

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     title = {Double ramification cycles on the moduli spaces of curves},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     pages = {221--266},
     publisher = {Springer Berlin Heidelberg},
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Janda, F.; Pandharipande, R.; Pixton, A.; Zvonkine, D. Double ramification cycles on the moduli spaces of curves. Publications Mathématiques de l'IHÉS, Tome 125 (2017), pp. 221-266. doi : 10.1007/s10240-017-0088-x. http://www.numdam.org/articles/10.1007/s10240-017-0088-x/

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