Supersingular curves of genus four in characteristic two

TL;DR

This paper investigates the structure of supersingular genus four curves in characteristic two, analyzing their Jacobians and stratifications to determine the dimension and properties of the supersingular locus.

Contribution

It characterizes the supersingular locus of Jacobians of genus four curves in characteristic two, showing it is pure of dimension three using geometric and group scheme analysis.

Findings

The supersingular Jacobian locus in characteristic two is pure of dimension three.

Analysis of Ekedahl-Oort types helps classify supersingular loci.

The study combines data on smooth genus four curves and abelian varieties over _2.

Abstract

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 supersingular locus in characteristic two. We show that the locus of supersingular Jacobians $\mathcal{S}_4\cap\mathcal{J}_4$ in characteristic two 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 $\mathbb{F}_2$, and another involves studying the Ekedahl-Oort types and using the indecomposability of some $\mathrm{BT}_1$ group schemes.