Ekedahl–Oort types of stable curves

We discuss an intrinsic definition of Ekedahl–Oort types for stable curves via Moonen’s Hasse–Witt triples, previously used for smooth curves. We show that this leads to an equivalent definition to the classical one for the Ekedahl–Oort types of generalized Jacobians as semi-abelian varieties. This approach enables us to compute bounds on the dimensions of many new Ekedahl–Oort loci of curves and to generalize results on the dimensions of the p-rank and a-number loci.