Pure and Applied Mathematics Quarterly

Volume 17 (2021)

Number 2

Special Issue: In Honor of David Mumford

Guest Editors: Ching-Li Chai, Amnon Neeman

Relations in the tautological ring of the moduli space of curves

Pages: 717 – 771

DOI: https://dx.doi.org/10.4310/PAMQ.2021.v17.n2.a7


R. Pandharipande (Departement Mathematik, Eidgenössische Technische Hochschule (ETH) Zürich, Switzerland)

A. Pixton (Department of Mathematics, University of Michigan, Ann Arbor, Mich., U.S.A.)


The virtual geometry of the moduli space of stable quotients is used to obtain Chow relations among the $\kappa$ classes on the moduli space of nonsingular genus $g$ curves. In a series of steps, the stable quotient relations are rewritten in successively simpler forms. The final result is the proof of the Faber–Zagier relations (conjectured in 2000).

Discussions with C. Faber played an important role in the development of our ideas. Much of the research reported here was done during visits of A.P. to IST Lisbon during the year 2010-11. R.P. was supported in Lisbon by a Marie Curie fellowship and a grant from the Gulbenkian foundation.

We thank the Forschungsinstitut für Mathematik at ETHZ for hosting several visits of A.P. in Zürich. R.P. was partially supported by SNF-200021143274, SNF-200020182181, ERC-2017-786580-MACI, SwissMAP, and the Einstein Stiftung. A.P. was supported by an NDSEG graduate fellowship and a fellowship from the Clay Mathematics Institute.

The project has received funding from the European Reseach Council (ERC) under the European Union Horizon 2020 Research and Innovation Program (grant No. 786580).

Received 19 September 2018

Accepted 12 January 2020

Published 12 May 2021