Journal article
An intersection-theoretic proof of the Harer–Zagier formula
A Giacchetto, D Lewański, P Norbury
Algebraic Geometry | EUROPEAN MATHEMATICAL SOC-EMS | Published : 2023
DOI: 10.14231/AG-2023-004
Abstract
We provide an intersection-theoretic formula for the Euler characteristic of the moduli space of smooth curves. This formula reads purely in terms of Hodge integrals, and, as a corollary, the standard calculus of tautological classes gives a new short proof of the Harer–Zagier formula. Our result is based on the Gauss–Bonnet formula, and on the observation that a certain parametrisation of the Ω-class – the Chern class of the universal rth root of the twisted log canonical bundle – provides the Chern class of the log tangent bundle to the moduli space of smooth curves. These Ω-classes have been recently employed in a great variety of enumerative problems. We produce a list of their propertie..
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Awarded by Horizon 2020 Framework Programme
Funding Acknowledgements
This journal is Foundation Compositio Mathematica 2023. This article is distributed with Open Access under the terms of the Creative Commons Attribution Non-Commercial License, which permits non-commercial reuse, distribution, and reproduction in any medium, provided that the original work is properly cited. For commercial re-use, please contact the Foundation Compositio Mathematica. This work is partly a result of the ERC-SyG project, Recursive and Exact New Quantum Theory (ReNewQuan-tum) which received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme under grant agreement No 810573. A.G. has been supported by the Max-Planck-Gesellschaft and the Institut de Physique Theorique (IPhT), CEA, Universite Paris-Saclay. D.L. has been supported by the section de Mathematiques de l'Universite de Geneve, by the Institut de Physique Theorique (IPhT) , CEA, Universite Paris-Saclay, by the Institut des Hautes Etudes Scientifiques (IHES) , Universite Paris-Saclay, and by the INdAM group GNSAGA. P.N. has been supported under the Australian Research Council Discovery Projects funding scheme project number DP180103891. Both A.G. and D.L. have been supported by the University of Melbourne who hosted the research visit which led to this collaboration.