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We describe the intersection of the Torelli locus $j(\mathcal{M}_4^{ct}) =\mathcal{J}_4$ with Newton and Ekedahl-Oort strata related to the supersingularlocus in characteristic two. We show that the locus of supersingular Jacobians$\mathcal{S}_4\cap\mathcal{J}_4$ in characteristic two is pure of dimensionthree. One way to obtain that result uses an analysis of the data of smoothgenus four curves and principally polarized abelian fourfolds defined over$\mathbb{F}_2$, and another involves studying the Ekedahl-Oort types and usingthe indecomposability of some $\mathrm{BT}_1$ group schemes.

Supersingular curves of genus four in characteristic two