Supersingular curves of genus four in characteristic two

We describe the intersection of the Torelli locus j ( M 4 c t ) = J 4 j(M_4ᶜᵗ) = 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 S_4∩ 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 F_2 , and another involves a more careful study of some relevant Ekedahl-Oort loci.