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arxiv: 2405.06341 · v2 · submitted 2024-05-10 · 🧮 math.AG

Finite symplectic automorphism groups of supersingular K3 surfaces

Hisanori Ohashi , Matthias Sch\"utt This is my paper

Pith reviewed 2026-05-24 01:02 UTC · model grok-4.3

classification 🧮 math.AG MSC 14J28
keywords K3 surfacessupersingular K3 surfacessymplectic automorphismsfinite groupsArtin invariantpositive characteristicalgebraic geometry
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The pith

Finite symplectic automorphism groups of supersingular K3 surfaces of Artin invariant one are completely classified.

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

The paper establishes a complete list of finite groups that can act symplectically on supersingular K3 surfaces with Artin invariant one. The classification is carried out directly in this special geometric setting. If the list is exhaustive, it supplies the full classification of all tame finite symplectic automorphism groups on arbitrary K3 surfaces together with the complete list of all finite symplectic automorphism groups on K3 surfaces in every characteristic greater than 11.

Core claim

Supersingular K3 surfaces of Artin invariant one admit only those finite symplectic automorphism groups that belong to an explicitly enumerated collection determined by the geometry of the surfaces in positive characteristic.

What carries the argument

The Artin invariant one condition, which forces the possible finite symplectic automorphism groups into a short explicit list.

If this is right

  • Tame finite symplectic automorphism groups on every K3 surface are now listed.
  • All finite symplectic automorphism groups on K3 surfaces in characteristic p greater than 11 are listed.
  • Only groups compatible with the supersingular lattice structure in characteristic p can arise.
  • The possible orders of such groups are bounded by the geometry of the Artin invariant one case.

Where Pith is reading between the lines

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

  • The same enumeration technique could be tested on supersingular K3 surfaces with higher Artin invariants to see whether new groups appear.
  • The classification supplies an upper bound on group orders that might be compared with known bounds coming from the K3 lattice in characteristic zero.
  • One could ask whether the listed groups realize every possible action or whether further geometric obstructions exist beyond the Artin invariant one case.

Load-bearing premise

The classification obtained for supersingular K3 surfaces of Artin invariant one extends, by reduction, to the general tame case on all K3 surfaces in characteristic greater than 11.

What would settle it

A concrete supersingular K3 surface of Artin invariant one equipped with a symplectic action by a finite group absent from the enumerated list would refute the classification.

Figures

Figures reproduced from arXiv: 2405.06341 by Hisanori Ohashi, Matthias Sch\"utt.

Figure 1
Figure 1. Figure 1: Primitive sublattices of L = L1,25 From [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: 32 smooth rational curves E ± i , ℓ± j on Y˜ with [20, [PITH_FULL_IMAGE:figures/full_fig_p039_2.png] view at source ↗
read the original abstract

We give a complete classification of finite groups acting symplectically on supersingular K3 surfaces of Artin invariant one. Using work of Dolgachev and Keum, this provides the full classification of tame finite symplectic automorphism groups on any K3 surface, and in particular of all finite symplectic automorphism groups on K3 surfaces in characteristic p>11.

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.

Referee Report

1 major / 0 minor

Summary. The manuscript claims to give a complete classification of finite groups acting symplectically on supersingular K3 surfaces of Artin invariant one. Using work of Dolgachev and Keum, this is asserted to yield the full classification of tame finite symplectic automorphism groups on arbitrary K3 surfaces and, in particular, all such groups in characteristic p>11.

Significance. If the classification of the supersingular Artin-invariant-one case is complete and the cited reduction applies without gaps, the result would furnish the definitive list of tame finite symplectic automorphism groups of K3 surfaces, extending known results to positive characteristic. This would be a substantial contribution to the study of automorphisms of K3 surfaces.

major comments (1)
  1. [Introduction / §1 (reduction step)] The extension of the supersingular Artin-invariant-one classification to the full tame case on arbitrary K3 surfaces (and thus to all p>11) depends entirely on the reduction theorem of Dolgachev and Keum. The manuscript must contain an explicit verification, in a dedicated section or subsection, that every tame symplectic action on a general K3 surface maps to an action on a supersingular surface of Artin invariant exactly one while preserving the group and the symplectic condition, with no exceptions for p>11; absent such a verification the completeness claim for the general case cannot be assessed.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting the need for explicit verification of the reduction step. We address the major comment below.

read point-by-point responses

  1. Referee: The extension of the supersingular Artin-invariant-one classification to the full tame case on arbitrary K3 surfaces (and thus to all p>11) depends entirely on the reduction theorem of Dolgachev and Keum. The manuscript must contain an explicit verification, in a dedicated section or subsection, that every tame symplectic action on a general K3 surface maps to an action on a supersingular surface of Artin invariant exactly one while preserving the group and the symplectic condition, with no exceptions for p>11; absent such a verification the completeness claim for the general case cannot be assessed.

    Authors: We agree that the manuscript would benefit from a self-contained verification of the Dolgachev-Keum reduction to make the completeness claim fully transparent. While the current text cites their work and asserts the extension, it does not include a dedicated outline of the reduction process. In the revised version we will add a new subsection (in §1) that explicitly verifies the applicability: it will recall the relevant statements from Dolgachev-Keum, confirm that every tame symplectic action on a general K3 surface reduces to an action on a supersingular K3 of Artin invariant exactly one, and check that the group and symplectic condition are preserved with no exceptions for p>11. This addition will be based directly on the cited reference and will not alter the classification results themselves. revision: yes

Circularity Check

0 steps flagged

No significant circularity; classification relies on external theorem

full rationale

The paper's core result is a classification of finite symplectic groups on supersingular K3 surfaces of Artin invariant 1. The extension to tame groups on arbitrary K3 surfaces (including p>11) is explicitly attributed to the external work of Dolgachev and Keum, who are distinct from the present authors. No self-citation load-bearing, self-definitional reduction, fitted-input prediction, or other enumerated circular pattern appears in the provided abstract or described claims. The derivation chain for the supersingular case does not reduce to its own inputs by construction, and the cited result is independent external support.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only the abstract is available, so the ledger records the background assumptions typical for the field rather than specific free parameters or new entities introduced in the proofs.

axioms (2)
  • standard math Standard definitions and properties of K3 surfaces, symplectic forms, and Artin invariants in algebraic geometry over fields of positive characteristic.
    The classification presupposes the usual framework of algebraic geometry for K3 surfaces.
  • domain assumption The cited results of Dolgachev and Keum on tame finite symplectic automorphism groups apply without additional restrictions in characteristic p>11.
    The abstract invokes this prior work to extend the classification.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. The 3-divisibility of divisors on K3 surfaces in characteristic 3

    math.AG 2026-04 unverdicted novelty 6.0

    Possible 3-divisible A2^n configurations of smooth rational curves on K3 surfaces in char 3 are described and the resulting triple covers are fully classified.

Reference graph

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