An intersection-theoretic proof of the Harer–Zagier formula

Giacchetto, Alessandro; Lewański, Danilo; Norbury, Paul

doi:10.14231/AG-2023-004

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An intersection-theoretic proof of the Harer–Zagier formula

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Giacchetto, Alessandro
Max Planck Institute for Mathematics, Max Planck Society;

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https://doi.org/10.14231/AG-2023-004
(Publisher version)

https://doi.org/10.48550/arXiv.2112.11137
(Preprint)

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Citation

Giacchetto, A., Lewański, D., & Norbury, P. (2023). An intersection-theoretic proof of the Harer–Zagier formula. Algebraic Geometry, 10(2), 130-147. doi:10.14231/AG-2023-004.

Cite as: https://hdl.handle.net/21.11116/0000-000C-CF0C-F

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 properties, proving new ones, collecting the properties already in the literature
or only known to the experts, and extending some of them.