A ball quotient parametrizing trigonal genus 4 curves

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 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 $g^1_3$.