Ekedahl Oort Strata Research Articles

Ekedahl-Oort strata in the supersingular locus

The Ekedahl–Oort stratification is a stratification of the moduli space of abelian varieties in characteristic p>0. We give a description of the individual Ekedahl–Oort strata contained in the supersingular locus in terms of Deligne–Lusztig varieties, refining a result of Harashita. We apply this description to calculate the number of components of a supersingular Ekedahl–Oort stratum.

Ekedahl-Oort strata contained in the supersingular locus and Deligne-Lusztig varieties

We study the moduli space of principally polarized abelian varieties over fields of positive characteristic. In this paper we describe certain unions of Ekedahl-Oort strata contained in the supersingular locus in terms of Deligne-Lusztig varieties. As a corollary we show that each Ekedahl-Oort stratum contained in the supersingular locus is reducible except possibly for small p p .

Ekedahl-Oort strata and the first Newton slope strata

We investigate stratifications on the moduli space A g \mathcal {A}_g of principally polarized abelian varieties of dimension g g in characteristic p > 0 p>0 . In this paper we give an easy algorithm determining the first Newton slope of any generic point of each Ekedahl-Oort stratum.

Discrete invariants of varieties in positive characteristic

If S is a scheme of characteristic p, we define an F-zip over S to be a vector bundle with two filtrations plus a collection of semilinear isomorphisms between the graded pieces of the filtrations. For every smooth proper morphism X → S satisfying certain conditions, the de Rham bundles HdRn(X/S) have a natural structure of an F-zip. We givea complete classification of F-zips over an algebraically closed field by studying a semilinear variant of a variety that appears in recent work of Lusztig. For every F-zip over S, our methods give a scheme-theoretic stratification of S. If the F-zip is associated to an abelian scheme over S, the underlying topological stratification is the Ekedahl-Oort stratification. We conclude the paper with a discussion of several examples, such as good reductions of Shimura varieties of PEL type and K3 surfaces.

A dimension formula for Ekedahl-Oort strata

We study the Ekedahl-Oort stratification on moduli spaces of PEL type. The strata are indexed by the classes in a Weyl group modulo a subgroup, and each class has a distinguished representative of minimal length. The main result of this paper is that the dimension of a stratum equals the length of the corresponding Weyl group element. We also discuss some explicit examples.