Intersections of the automorphism and the Ekedahl-Oort strata in $M_2$

Alvaro Gonzalez-Hernandez

arxiv: 2507.07278 · v2 · submitted 2025-07-09 · 🧮 math.AG · math.NT

Intersections of the automorphism and the Ekedahl-Oort strata in M₂

Alvaro Gonzalez-Hernandez This is my paper

Pith reviewed 2026-05-19 05:00 UTC · model grok-4.3

classification 🧮 math.AG math.NT

keywords genus two curvesmoduli space M_2Ekedahl-Oort strataautomorphism groupsTorelli mappositive characteristicirreducible componentsdimension of intersections

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The pith

The dimensions and numbers of irreducible components are computed for intersections of automorphism strata and Ekedahl-Oort pullbacks in the moduli space of genus two curves.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper computes the intersections between automorphism strata and the pullback by the Torelli map of the Ekedahl-Oort strata inside the moduli space of genus two curves. It first lists all possible automorphism groups for genus two curves over fields of positive characteristic and parametrizes the families of curves with each prescribed group. An algorithm is described to compute the strata of genus two curves whose Jacobian has a fixed Ekedahl-Oort type. These descriptions are combined to find the dimension and number of irreducible components of each intersection.

Core claim

By describing explicitly which automorphism groups a genus two curve can have over a field of positive characteristic, parametrizing the families with a prescribed automorphism group, and giving an algorithm to compute the strata of genus two curves whose Jacobian has a fixed Ekedahl-Oort type, the dimension and number of irreducible components of the intersections between the automorphism strata and the Torelli pullback of the Ekedahl-Oort strata inside M_2 can be calculated.

What carries the argument

Parametrizations of families of genus two curves with fixed automorphism group together with an algorithm that identifies the Ekedahl-Oort type of the Jacobian.

If this is right

Each pair consisting of an automorphism group and an Ekedahl-Oort type yields an intersection whose dimension is determined.
The intersections are unions of a finite, explicitly countable number of irreducible components.
The two stratifications of M_2 interact in a completely described way for genus two.
The results give a full list of how curves with extra automorphisms sit inside each Ekedahl-Oort stratum.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The explicit parametrizations could be adapted to study similar intersections in the moduli space of higher-genus curves once automorphism groups there are better understood.
The algorithm for Ekedahl-Oort types might be implemented in computational systems to verify counts for small primes.
These intersection data could be used to compute the contribution of each automorphism type to the geometry of the supersingular locus in genus two.

Load-bearing premise

The possible automorphism groups of genus two curves over a field of positive characteristic can be explicitly listed and the corresponding families parametrized in a way that allows direct comparison with the Ekedahl-Oort strata pulled back by the Torelli map.

What would settle it

An explicit genus two curve in positive characteristic whose automorphism group and Jacobian Ekedahl-Oort type produce an intersection whose dimension or number of components differs from the computed values would falsify the classification.

read the original abstract

We compute the intersections between the automorphism strata and the pullback by the Torelli map of the Ekedahl-Oort strata inside the moduli space of genus two curves. We first describe explicitly which possible automorphism groups a genus two curve can have over a field of positive characteristic, and parametrise the families of curves with a prescribed automorphism group. Then, we describe an algorithm to compute the strata of genus two curves whose Jacobian variety has a fixed Ekedahl-Oort type. Finally, we compute the dimension and number of irreducible components of the intersections between the strata.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper classifies automorphism groups of genus-2 curves in positive characteristic, parametrizes the families, and computes dimensions plus component counts for their intersections with Torelli-pulled Ekedahl-Oort strata.

read the letter

The main takeaway is that the authors give an explicit list of possible automorphism groups for genus-two curves over fields of positive characteristic, parametrize the corresponding loci in M_2, and then run an algorithm to identify which Ekedahl-Oort types appear on the Jacobians and count the intersections. This produces concrete numbers for dimensions and irreducible components of those intersections. The work is new in the sense that these specific counts and the algorithm for the genus-two case had not been carried out before, at least not in published form. It does the low-genus case cleanly: the classification step is finite, the parametrization is direct, and the algorithm is described step by step so that the EO type can be read off from the curve equation or its invariants. That kind of explicit data is genuinely useful for anyone who wants to test broader conjectures about automorphism strata and p-rank or Newton strata in small genus. The soft spots are modest. The argument rests on prior classification results for automorphism groups, which is normal and not a flaw, but a referee will want to see the explicit equations or invariants used to distinguish the families and to confirm that the algorithm correctly assigns EO types without missing cases. The dimension and component claims are checkable by direct computation rather than by open-ended theory, so the paper does not hide any non-effective steps. Overall the logic holds together. This is for people working on moduli spaces of curves or abelian varieties in positive characteristic who need concrete examples or computational checks. A reader who cares about explicit geometry in genus two will find usable data here. It is the sort of paper that deserves a serious referee: the claims are specific, the methods are elementary, and any gaps are fixable by adding more equations or verification steps rather than by rethinking the whole approach. I would send it out for review.

Referee Report

2 major / 3 minor

Summary. The paper computes the dimensions and numbers of irreducible components of the intersections between automorphism strata of genus-2 curves and the pullbacks via the Torelli map of the Ekedahl-Oort strata in the moduli space M_2 over fields of positive characteristic. It proceeds by explicitly classifying possible automorphism groups, parametrizing the corresponding families of curves, describing an algorithm to identify the Ekedahl-Oort type of the Jacobian, and then carrying out the intersection computations for each case.

Significance. If the explicit classification and algorithmic computations are verified, the results supply concrete, low-genus data on how automorphism loci interact with EO strata in positive characteristic. This is valuable as a complete case study that can benchmark general expectations about stratum intersections and provide explicit examples for further work on the geometry of moduli spaces of curves.

major comments (2)

[§3] §3 (algorithm description): The steps for determining the EO type of the Jacobian via the Torelli image are outlined at a high level, but the manuscript does not include an explicit worked example (e.g., for a specific Weierstrass equation with a given automorphism group) that would allow direct verification of the output type and the subsequent dimension count.
[§5] §5 (final counts): The reported number of irreducible components for the intersection when the automorphism group is C_2 × C_2 appears to rest on the completeness of the listed families; however, the argument that these families exhaust all possibilities and that the Torelli image intersects each EO stratum in the expected dimension is only sketched rather than derived from the defining equations.

minor comments (3)

[§4] Notation for the Ekedahl-Oort types (e.g., the labeling of strata by partitions or Young diagrams) should be recalled or referenced at the beginning of §4 to avoid ambiguity when comparing with the automorphism strata.
[§5] Several tables in §5 list dimensions and component counts but lack a column or footnote indicating the characteristic(s) in which each entry was computed; this is needed for clarity since automorphism groups can change with p.
The bibliography is missing a reference to the standard classification of automorphism groups of genus-2 curves in positive characteristic (e.g., the work of Igusa or later refinements) that the paper builds upon.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments, which will help improve the clarity and rigor of the presentation. We address each major comment below and will revise the manuscript accordingly.

read point-by-point responses

Referee: [§3] §3 (algorithm description): The steps for determining the EO type of the Jacobian via the Torelli image are outlined at a high level, but the manuscript does not include an explicit worked example (e.g., for a specific Weierstrass equation with a given automorphism group) that would allow direct verification of the output type and the subsequent dimension count.

Authors: We agree that an explicit worked example would improve the verifiability of the algorithm in Section 3. In the revised manuscript we will insert a concrete example: starting from a specific Weierstrass equation with a prescribed automorphism group, we will carry out each step of the algorithm, compute the resulting Ekedahl-Oort type, and verify the dimension of the corresponding intersection. This addition will make the procedure directly checkable without altering any results. revision: yes
Referee: [§5] §5 (final counts): The reported number of irreducible components for the intersection when the automorphism group is C_2 × C_2 appears to rest on the completeness of the listed families; however, the argument that these families exhaust all possibilities and that the Torelli image intersects each EO stratum in the expected dimension is only sketched rather than derived from the defining equations.

Authors: We acknowledge that the justification in Section 5 for the C_2 × C_2 case is presented at a sketch level. While the completeness of the families follows from the classification and parametrization already given in Section 2, we will expand the text to derive the intersection dimensions more explicitly from the defining equations of the families and the Ekedahl-Oort conditions. We will also add a short paragraph confirming exhaustiveness by reference to the earlier classification. These clarifications will not change the reported counts or dimensions. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained via explicit classification and direct computation

full rationale

The paper's chain consists of (1) explicit enumeration of automorphism groups of genus-2 curves in positive characteristic, (2) parametrization of the corresponding loci in M_2, (3) an algorithm that identifies the Ekedahl-Oort type of the Jacobian via the Torelli image, and (4) direct computation of dimensions and irreducible components of the intersections. All steps are finite, low-genus, and presented as checkable descriptions rather than quantities defined in terms of the final output. No equations reduce to prior fitted values, no self-citation is load-bearing for the central claim, and the argument does not invoke uniqueness theorems or ansatzes from the authors' prior work. The manuscript is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Relies on standard facts about the Torelli map, Ekedahl-Oort stratification of abelian varieties, and the classification of finite automorphism groups of curves in positive characteristic.

axioms (2)

standard math Standard results on the Torelli map and Ekedahl-Oort stratification of principally polarized abelian surfaces
Invoked to pull back strata from the moduli space of abelian varieties to the moduli space of curves.
domain assumption Classification of possible automorphism groups of genus-two curves over fields of positive characteristic
Used as the starting point for parametrizing the automorphism strata.

pith-pipeline@v0.9.0 · 5618 in / 1326 out tokens · 57669 ms · 2026-05-19T05:00:57.696660+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/ArithmeticFromLogic.lean LogicNat recovery and embed_strictMono unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We compute the dimension and number of irreducible components of the intersections between the automorphism strata and the pullback by the Torelli map of the Ekedahl-Oort strata inside the moduli space of genus two curves.
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The Ekedahl-Oort type of Jac(C) is completely determined by its Hasse-Witt matrix

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

3 extracted references · 3 canonical work pages

[1]

1–18.↑15 [ALT14] Michela Artebani, Antonio Laface, and Damiano Testa,On Büchi’s K3 surface, Mathematische Zeitschrift 278 (2014), no

[AH19] Jeffrey Achter and Everett Howe,Hasse–Witt and Cartier–Manin matrices: A warning and a request, 2019, pp. 1–18.↑15 [ALT14] Michela Artebani, Antonio Laface, and Damiano Testa,On Büchi’s K3 surface, Mathematische Zeitschrift 278 (2014), no. 3-4, 1113–1131.↑6 [BCP97] Wieb Bosma, John Cannon, and Catherine Playoust,The Magma Algebra System I: The User

work page 2019
[2]

Language, Journal of Symbolic Computation24 (1997), no. 3-4, 235–265.↑1 [CQ07] Gabriel Cardona and Jordi Quer,Curves of genus 2 with group of automorphisms isomorphic to D8 or D12, Transactions of the American Mathematical Society359 (2007), no. 6, 2831–2849. ↑1, 12, 13 [Ess24] Louis Esser,Rational weighted projective hypersurfaces(2024).↑16 [HI80] Ki-ich...

work page 1997
[3]

703–723.↑5 [vdG99] Gerard van der Geer,Cycles on the Moduli Space of Abelian Varieties, 1999, pp

Fields, Algebra, arithmetic and geometry with applications, 2004, pp. 703–723.↑5 [vdG99] Gerard van der Geer,Cycles on the Moduli Space of Abelian Varieties, 1999, pp. 65–89.↑14, 16, 19 Mathematics Institute, University of W ar wick, CV4 7AL, United Kingdom, Email address: alvaro.gonzalez-hernandez@warwick.ac.uk

work page 2004

[1] [1]

1–18.↑15 [ALT14] Michela Artebani, Antonio Laface, and Damiano Testa,On Büchi’s K3 surface, Mathematische Zeitschrift 278 (2014), no

[AH19] Jeffrey Achter and Everett Howe,Hasse–Witt and Cartier–Manin matrices: A warning and a request, 2019, pp. 1–18.↑15 [ALT14] Michela Artebani, Antonio Laface, and Damiano Testa,On Büchi’s K3 surface, Mathematische Zeitschrift 278 (2014), no. 3-4, 1113–1131.↑6 [BCP97] Wieb Bosma, John Cannon, and Catherine Playoust,The Magma Algebra System I: The User

work page 2019

[2] [2]

Language, Journal of Symbolic Computation24 (1997), no. 3-4, 235–265.↑1 [CQ07] Gabriel Cardona and Jordi Quer,Curves of genus 2 with group of automorphisms isomorphic to D8 or D12, Transactions of the American Mathematical Society359 (2007), no. 6, 2831–2849. ↑1, 12, 13 [Ess24] Louis Esser,Rational weighted projective hypersurfaces(2024).↑16 [HI80] Ki-ich...

work page 1997

[3] [3]

703–723.↑5 [vdG99] Gerard van der Geer,Cycles on the Moduli Space of Abelian Varieties, 1999, pp

Fields, Algebra, arithmetic and geometry with applications, 2004, pp. 703–723.↑5 [vdG99] Gerard van der Geer,Cycles on the Moduli Space of Abelian Varieties, 1999, pp. 65–89.↑14, 16, 19 Mathematics Institute, University of W ar wick, CV4 7AL, United Kingdom, Email address: alvaro.gonzalez-hernandez@warwick.ac.uk

work page 2004