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A BALL QUOTIENT PARAMETRIZING TRIGONAL GENUS 4 CURVES

Part of: Complex spaces with a group of automorphisms Automorphic functions Curves

Published online by Cambridge University Press:  21 September 2023

EDUARD LOOIJENGA Open the ORCID record for EDUARD LOOIJENGA [Opens in a new window]
EDUARD LOOIJENGA*
Affiliation:
Department of Mathematics University of Chicago Chicago, Illinois 60637 USA and Universiteit Utrecht Utrecht Netherlands
*
e.j.n.looijenga@uu.nl
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Abstract

We consider the moduli space of genus 4 curves endowed with a $g^1_3$ (which maps with degree 2 onto the moduli space of genus 4 curves). We prove that it defines a degree $\frac {1}{2}(3^{10}-1)$ cover of the nine-dimensional Deligne–Mostow ball quotient such that the natural divisors that live on that moduli space become totally geodesic (their normalizations are eight-dimensional ball quotients). This isomorphism differs from the one considered by S. Kondō, and its construction is perhaps more elementary, as it does not involve K3 surfaces and their Torelli theorem: the Deligne–Mostow ball quotient parametrizes certain cyclic covers of degree 6 of a projective line and we show how a level structure on such a cover produces a degree 3 cover of that line with the same discriminant, yielding a genus 4 curve endowed with a $g^1_3$.

MSC classification

Primary: 14H30: Coverings, fundamental group 32M15: Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras
Type
Article
Information
Nagoya Mathematical Journal , Volume 254 , June 2024 , pp. 366 - 378
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Foundation Nagoya Mathematical Journal

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Footnotes

The author was supported by the Jump Trading Mathlab Research Fund.

References

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