Supersingular curves of genus four in characteristic two

Abstract

We describe the intersection of the Torelli locus j ( M 4 c t ) = J 4 j(\mathcal {M}_4^{ct}) = \mathcal {J}_4 with Newton and Ekedahl-Oort strata related to the supersingular locus in characteristic 2. We show that the locus of supersingular Jacobians S 4 ∩ J 4 \mathcal {S}_4\cap \mathcal {J}_4 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 F 2 \mathbb {F}_2 , and another involves a more careful study of some relevant Ekedahl-Oort loci.