Ekedahl–Oort strata on the moduli space of curves of genus four

Abstract

We study the induced Ekedahl–Oort stratification on the moduli of curves of genus 4 in positive characteristic by computing the de Rham cohomology of curves. For moduli space of hyperelliptic curves of genus 4 in characteristic 3, we determine the dimension and reducibility of Ekedahl–Oort strata. For moduli space of curves of genus 4 in odd characteristic, we show the existence of certain Ekedahl–Oort strata and discuss the existence of superspecial curves.