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Ekedahl-Oort types of stable curves

Dušan Dragutinović

TL;DR

The paper extends Ekedahl-Oort (EO) theory to stable curves by using Moonen's Hasse-Witt triple framework, proving that the EO type of a stable curve μ(C) agrees with that of its generalized Jacobian μ(J_C) and with the EO type of its normalization μ(ṫC). This intrinsic description enables an inductive method to bound the dimensions of EO loci for curves, including loci defined by p-rank and a-number constraints, and to transfer results from abelian varieties to the moduli of curves via the boundary Δ_0. The authors derive a general bound (and special cases) for the codimension of EO loci in ar{M}_g, provide applications to hyperelliptic genus-4 curves, and analyze EO stratifications in characteristics p = 2 and p = 3, including explicit examples and a Fibonacci-type count of possible final types. Overall, the work unifies EO stratifications across abelian and stable-curve settings, illuminates boundary phenomena, and yields practical dimension estimates for EO, p-rank, and a-number loci in moduli spaces of 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.

Ekedahl-Oort types of stable curves

TL;DR

The paper extends Ekedahl-Oort (EO) theory to stable curves by using Moonen's Hasse-Witt triple framework, proving that the EO type of a stable curve μ(C) agrees with that of its generalized Jacobian μ(J_C) and with the EO type of its normalization μ(ṫC). This intrinsic description enables an inductive method to bound the dimensions of EO loci for curves, including loci defined by p-rank and a-number constraints, and to transfer results from abelian varieties to the moduli of curves via the boundary Δ_0. The authors derive a general bound (and special cases) for the codimension of EO loci in ar{M}_g, provide applications to hyperelliptic genus-4 curves, and analyze EO stratifications in characteristics p = 2 and p = 3, including explicit examples and a Fibonacci-type count of possible final types. Overall, the work unifies EO stratifications across abelian and stable-curve settings, illuminates boundary phenomena, and yields practical dimension estimates for EO, p-rank, and a-number loci in moduli spaces of 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 -rank and -number loci of curves.
Paper Structure (12 sections, 19 theorems, 44 equations)

This paper contains 12 sections, 19 theorems, 44 equations.

Table of Contents

  1. Introduction
  2. Ekedahl-Oort types of abelian varieties
  3. Ekedahl-Oort types of smooth curves
  4. On the Weyl group and the inclusion of Ekedahl-Oort loci
  5. Ekedahl-Oort types of stable curves
  6. Inductive results
  7. Inductive results about the Ekedahl-Oort strata
  8. Some applications
  9. On Ekedahl-Oort loci of smooth hyperelliptic curves of genus $4$
  10. Examples in characteristic $p = 2$ and $p = 3$
  11. Ekedahl-Oort stratification of $\mathcal{M}_4$ in characteristic $2$
  12. Ekedahl-Oort stratification of $\mathcal{M}_g$ in characteristic $3$

Key Result

Theorem A

Let $C$ be a stable curve, $\mathcal{J}_C$ its generalized Jacobian, and $h: \Tilde{C} \to C$ its normalization. Then $\mu(\mathcal{J}_C) = \mu(C) = \mu(\Tilde{C})$, where $\mu(\mathcal{J}_C)$ is classically defined, while $\mu(C)$ and $\mu(\Tilde{C})$ are defined in terms of Hasse-Witt triples.

Theorems & Definitions (47)

  • Theorem A: Theorem \ref{['thm:muCnorm']}
  • Theorem B: Theorem \ref{['thm:main1']}
  • Example 2.1
  • Theorem 2.2: moonen
  • Lemma 3.1
  • proof
  • Definition 3.2
  • Definition 3.3
  • Theorem 3.4
  • proof
  • ...and 37 more