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arxiv: 2605.02522 · v1 · submitted 2026-05-04 · 🧮 math.AG · math.RT

Deligne-Lusztig varieties whose canonical divisors have negativity

Ulrich G\"ortz , Stefan Schr\"oer This is my paper

Pith reviewed 2026-05-08 17:53 UTC · model grok-4.3

classification 🧮 math.AG math.RT
keywords Deligne-Lusztig varietiescanonical divisorssupersingular K3 surfacesArtin invariantRee curveIsogeny TheoremSuzuki-Ree casesTutte-Coxeter graph
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The pith

Compactified Deligne-Lusztig varieties with negative canonical divisors explicitly realize supersingular K3 surfaces and Ree curves in small characteristics.

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

The paper examines compactified Deligne-Lusztig varieties where the canonical divisor, expressed as a linear combination of boundary divisors, has all coefficients strictly negative or zero. In dimension two the case of type C2 produces the supersingular K3 surface with Artin invariant sigma equals one in characteristic two. The twisted cases of types 2A2, 2C2, and 2G2 produce ruled surfaces canonically attached to supersingular elliptic curves in characteristic two, or the Ree curve of genus fifteen in characteristic three. To reach these descriptions the authors introduce a general framework for Deligne-Lusztig varieties that relies on the Isogeny Theorem and works directly over prime fields, naturally including the Suzuki-Ree cases.

Core claim

We investigate compactified Deligne-Lusztig varieties whose canonical divisor, when expressed as a linear combination of boundary divisors, has all coefficients strictly negative or zero. In dimension two we obtain explicit descriptions: the case C2 gives the supersingular K3 surface with Artin invariant sigma=1 in characteristic two. The twisted cases 2A2 and 2C2 and 2G2 yield particular ruled surfaces, attached in a canonical way to supersingular elliptic curves in characteristic two, or the Ree curve in characteristic three. To obtain such results we develop a new general framework for Deligne-Lusztig varieties that relies on the Isogeny Theorem and works over prime fields rather than ихr

What carries the argument

A new general framework for Deligne-Lusztig varieties that relies on the Isogeny Theorem, works over prime fields rather than algebraic closures, and includes the Suzuki-Ree cases without effort.

Load-bearing premise

The arguments depend on properties of the Tutte-Coxeter graph and on the Isogeny Theorem applying directly over prime fields rather than algebraic closures.

What would settle it

A computation showing that the compactified Deligne-Lusztig variety of type C2 in characteristic two is not isomorphic to the supersingular K3 surface with Artin invariant one, or that its canonical divisor has a positive coefficient.

Figures

Figures reproduced from arXiv: 2605.02522 by Stefan Schr\"oer, Ulrich G\"ortz.

Figure 1
Figure 1. Figure 1: The Tutte–Coxeter graph irreducible components of the divisors X¯(s1) and X¯(s2) form an snc configuration of projective lines whose dual graph stems from the building for the symplectic group G = Sp4 , cf. Proposition 1.23. This is depicted in view at source ↗
read the original abstract

We investigate compactified Deligne-Lusztig varieties whose canonical divisor, when expressed as a linear combination of boundary divisors, has all coefficients strictly negative or zero. In dimension two we obtain explicit descriptions: The case $C_2$ gives the supersingular K3 surface with Artin invariant $\sigma=1$ in characteristic two. The arguments rely on properties of the Tutte-Coxeter graph; in this connection we also gain some insight into the arithmetic of quasi-elliptic Weierstrass equations and rational double points. The twisted cases ${}^2A_2$ and ${}^2C_2$ and ${}^2G_2$ yield particular ruled surfaces, attached in a canonical way to supersingular elliptic curves in characteristic two, or the Ree curve in characteristic three. The latter is a curve of genus fifteen with outstanding symmetries. In the former cases, the surfaces can also be expressed as symmetric squares. To obtain such results, we develop a new general framework for Deligne-Lusztig varieties that relies on the Isogeny Theorem, and works over prime fields rather than their algebraic closures, and includes the notorious Suzuki-Ree cases without effort.

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

0 major / 2 minor

Summary. The paper studies compactified Deligne-Lusztig varieties whose canonical divisor, when expressed as a linear combination of boundary divisors, has all coefficients strictly negative or zero. In dimension two it supplies explicit descriptions: the C_2 case is identified with the supersingular K3 surface of Artin invariant σ=1 in characteristic two; the twisted cases ^2A_2, ^2C_2 and ^2G_2 yield ruled surfaces canonically attached to supersingular elliptic curves (char. 2) or the Ree curve of genus 15 (char. 3). The arguments rely on properties of the Tutte-Coxeter graph and on a new general framework that invokes the Isogeny Theorem over prime fields rather than algebraic closures and accommodates the Suzuki-Ree cases without additional effort. Additional arithmetic insights are obtained for quasi-elliptic Weierstrass equations and rational double points.

Significance. If the identifications hold, the work supplies concrete geometric models linking Deligne-Lusztig varieties to supersingular K3 surfaces and to curves of exceptional symmetry, while the new Isogeny-Theorem framework operating directly over prime fields constitutes a methodological advance that may simplify future computations in positive-characteristic algebraic geometry. The combinatorial input from the Tutte-Coxeter graph yields independent arithmetic information on Weierstrass models and rational double points.

minor comments (2)
  1. The abstract states that the surfaces in the ^2A_2 and ^2C_2 cases 'can also be expressed as symmetric squares'; a brief indication of the precise construction (or a reference to the relevant proposition) would help the reader connect this description to the ruled-surface realization.
  2. The reliance on the Tutte-Coxeter graph is invoked for both the C_2 identification and the arithmetic insights; a short subsection summarizing the precise graph-theoretic properties used (e.g., which subgraphs or automorphisms appear) would improve readability for readers less familiar with the graph.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of our manuscript, including the accurate summary of our results on compactified Deligne-Lusztig varieties and the recognition of the methodological contribution of the Isogeny Theorem framework over prime fields. We appreciate the recommendation for minor revision and the acknowledgment of the links to supersingular K3 surfaces and curves of exceptional symmetry such as the Ree curve.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper constructs a new framework for Deligne-Lusztig varieties over prime fields using the external Isogeny Theorem and combinatorial properties of the Tutte-Coxeter graph. The explicit geometric identifications (e.g., the C2 case yielding the supersingular K3 surface with Artin invariant 1) are presented as derived outcomes rather than inputs or self-definitions. No load-bearing steps reduce by construction to fitted parameters, self-citations, or renamed known results; the derivation remains independent of the target claims and relies on externally verifiable tools.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based solely on the abstract; no free parameters or invented entities are introduced. The work relies on standard mathematical theorems and domain-specific combinatorial objects.

axioms (2)
  • domain assumption Properties of the Tutte-Coxeter graph
    Invoked for the C2 case arguments.
  • standard math Isogeny Theorem
    Used as the foundation for the new general framework over prime fields.

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