Computing discrete invariants of varieties in positive characteristic I. Ekedahl-Oort types of curves

Abstract

We develop a method to compute the Ekedahl–Oort type of a curve C over a field k of characteristic p (which is the isomorphism type of the p-kernel group scheme J[p], where J is the Jacobian of C). Part of our method is general, in that we introduce the new notion of a Hasse–Witt triple, which re-encodes in a useful way the information contained in the Dieudonné module of J[p]. For complete intersection curves we then give a simple method to compute this Hasse–Witt triple. An implementation of this method is available in Magma.