Supersingular curves of genus four in characteristic two

We describe the intersection of the Torelli locus j (M 4 c t) = J 4 j (M₄^ct) = J₄ 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 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 F 2 F₂, and another involves a more careful study of some relevant Ekedahl-Oort loci.