Supersingular Curves of Genus Four in Characteristic Two

Abstract

We describe the intersection of the Torelli locus j(Mct4 )=J4 with Newton and Ekedahl-Oort strata related to the supersingular locus in characteristic 2. We show that the locus of supersingular Jacobians S4 ∩ J4 in characteristic 2 is pure of dimension three. One way to obtain that result uses an analysis of the data of smooth genus four curves and principally polarized abelian fourfolds defined over F2, and another involves a more careful study of some relevant Ekedahl-Oort loci.