A ball quotient parametrizing trigonal genus 4 curves

DOI10.1017/NMJ.2023.28arXiv2211.09941OpenAlexW4386931156MaRDI QIDQ6417650

Eduard Looijenga

Publication date: 17 November 2022

Abstract: We consider the moduli space of genus 4 curves endowed with a $g_3^1$ (which maps with degree 2 onto the moduli space of genus 4 curves). We prove that it defines a degree $frac12(3^{10}-1)$ cover of the 9-dimensional Deligne-Mostow ball quotient such that the natural divisors that live on that moduli space become totally geodesic (their normalizations are 8-dimensional ball quotients). This isomorphism differs from the one considered by S. Kond=o 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 $g_3^1$.

Full work available at URL: https://doi.org/10.1017/nmj.2023.28

Mathematics Subject Classification ID

Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects) (32M15) Coverings of curves, fundamental group (14H30)

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