Ekedahl-Oort types of stable curves

TL;DR

This paper generalizes the concept of Ekedahl-Oort types from smooth to all stable curves, linking them to generalized Jacobians, and uses this to compute dimensions of related loci, extending prior results.

Contribution

It extends the definition of Ekedahl-Oort types to all stable curves and connects them to generalized Jacobians, enabling dimension calculations of specific loci.

Findings

Computed dimensions of Ekedahl-Oort loci of curves.

Generalized results on p-rank and a-number loci.

Unified the definitions for smooth and stable curves.

Abstract

We extend Moonen's definition of Ekedahl-Oort types of smooth curves in terms of Hasse-Witt triples to all stable curves and show that it matches Ekedahl and van der Geer's definition of Ekedahl-Oort types of their generalized Jacobians as semi-abelian varieties. Using this intrinsic insight, we can compute the dimensions of certain Ekedahl-Oort loci of curves and generalize some previously known results about the dimensions of the $p$-rank and $a$-number loci of curves.