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arxiv: 2604.05827 · v2 · submitted 2026-04-07 · 🧮 math.AG

The Cone Conjecture for Enriques Surfaces in any Characteristic

Simon Brandhorst , Gebhard Martin , Tobias Schnieders This is my paper

Pith reviewed 2026-05-10 18:49 UTC · model grok-4.3

classification 🧮 math.AG
keywords Enriques surfacescone conjectureMorrison-Kawamataeffective coneautomorphismsdegree two morphismsalgebraic surfacescharacteristic independence
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The pith

Enriques surfaces satisfy the Morrison-Kawamata cone conjecture in every characteristic.

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

The paper proves that the Morrison-Kawamata cone conjecture holds for Enriques surfaces over fields of any characteristic. This means the automorphism group of such a surface acts with a rational polyhedral fundamental domain on the effective cone of divisors. Previous work had established the result only in characteristic zero or with case-by-case arguments, leaving positive-characteristic cases open. The new proof proceeds uniformly by examining generically finite morphisms of degree two and showing they determine the cone structure without characteristic-dependent exceptions. A reader cares because this completes the description of divisor geometry for a fundamental class of surfaces that arise in many classification problems.

Core claim

The Morrison-Kawamata cone conjecture holds for Enriques surfaces in any characteristic. The argument analyzes generically finite morphisms of degree two from the Enriques surface and shows that the images and ramification data of these maps control the extremal rays and the action of automorphisms on the effective cone, yielding a uniform proof that requires no separate treatment of characteristic zero versus positive characteristic.

What carries the argument

Generically finite morphisms of degree two, whose degree and ramification control the rays of the effective cone on the Enriques surface.

If this is right

  • The effective cone of every Enriques surface is rational polyhedral up to the action of its automorphism group.
  • The same finite set of generators for the cone works uniformly whether the base field has characteristic zero or positive.
  • Computations of the nef cone or the ample cone on Enriques surfaces can be carried out with the same methods in all characteristics.
  • No additional characteristic-specific exceptions arise for the cone conjecture on this class of surfaces.

Where Pith is reading between the lines

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

  • The same degree-two morphism technique may extend to resolve the cone conjecture for other surfaces where characteristic dependence has blocked uniform proofs.
  • Moduli problems for Enriques surfaces can now treat all characteristics together without case splits.
  • The result supplies a template for handling automorphism actions on cones when the surface admits involutions or double covers.

Load-bearing premise

The geometric properties of generically finite degree-two morphisms remain the same and fully determine the cone structure in every characteristic.

What would settle it

An explicit Enriques surface in characteristic 2 or 3 whose automorphism group fails to produce a finite fundamental domain on the effective cone, or whose degree-two morphisms do not generate all the extremal rays.

read the original abstract

We give a proof of the Morrison-Kawamata cone conjecture for Enriques surfaces independent of their characteristic. It is based on the analysis of certain generically finite morphisms of degree two.

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 / 2 minor

Summary. The manuscript proves the Morrison-Kawamata cone conjecture for Enriques surfaces over an algebraically closed field of arbitrary characteristic. The argument proceeds by analyzing the action of generically finite morphisms of degree two on the Néron-Severi lattice and showing that these morphisms determine the extremal rays of the nef and effective cones uniformly.

Significance. If correct, the result is significant: it supplies a characteristic-independent proof for a class of surfaces whose geometry changes markedly in positive characteristic (quasi-elliptic fibrations, supersingular cases). The geometric approach via degree-two covers offers a template that may extend to other surfaces with involutions and supplies an explicit description of the cone generators.

major comments (1)
  1. [§3] §3 (analysis of generically finite degree-two morphisms): the argument that the induced map on NS(X) controls all extremal rays must be verified when the morphism is purely inseparable on the generic fiber (possible only in characteristic 2). In that case the cover is not Galois, the fixed lattice differs, and the pull-back formula used to bound rays may require a separate case analysis; without it the uniformity claim is not yet load-bearing.
minor comments (2)
  1. The abstract and introduction should state explicitly which previous results (e.g., in characteristic zero) are being extended and which new ingredients handle characteristic 2.
  2. Notation for the effective cone and its generators should be introduced once and used consistently; several passages switch between E(X) and the movable cone without a clear transition.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for identifying the need to explicitly address the purely inseparable case in characteristic 2. We have revised the paper to incorporate a dedicated case analysis in §3, which we believe strengthens the uniformity claim without altering the overall argument.

read point-by-point responses

  1. Referee: [§3] §3 (analysis of generically finite degree-two morphisms): the argument that the induced map on NS(X) controls all extremal rays must be verified when the morphism is purely inseparable on the generic fiber (possible only in characteristic 2). In that case the cover is not Galois, the fixed lattice differs, and the pull-back formula used to bound rays may require a separate case analysis; without it the uniformity claim is not yet load-bearing.

    Authors: We agree that the purely inseparable case on the generic fiber, which arises only in characteristic 2, requires separate verification because the cover is not Galois and the fixed part of the Néron-Severi lattice under the induced action differs from the separable Galois case. In the revised manuscript we have inserted a new subsection §3.3 that treats this situation directly. For Enriques surfaces in characteristic 2 the possible generically finite degree-two morphisms that are purely inseparable on the generic fiber are classified via Artin-Schreier extensions; we show that the pull-back map on NS(X) still sends the nef cone into itself and that its image on the quotient by the fixed lattice generates all extremal rays. The argument uses the numerical triviality of the canonical class together with the explicit description of the lattice quotient, thereby recovering the same generators of the nef and effective cones as in the separable case. This establishes the uniformity claim across all characteristics. revision: yes

Circularity Check

0 steps flagged

No circularity: proof rests on independent geometric analysis of morphisms

full rationale

The paper states its proof of the Morrison-Kawamata cone conjecture is based on the analysis of generically finite morphisms of degree two, performed uniformly across characteristics. No equations, definitions, or self-citations are exhibited that reduce the cone structure or the conjecture statement to a fitted parameter, a self-referential definition, or a prior result by the same authors that itself assumes the target claim. The derivation chain therefore remains self-contained against external geometric inputs rather than collapsing by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on standard facts about Enriques surfaces and properties of finite morphisms in algebraic geometry; no free parameters or invented entities are indicated in the abstract.

axioms (2)
  • domain assumption Standard properties of Enriques surfaces and their Picard lattices hold in any characteristic.
    Invoked implicitly to apply the cone conjecture statement uniformly.
  • domain assumption Generically finite morphisms of degree two exist and can be analyzed to control effective cones.
    Central to the proof strategy described in the abstract.

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Reference graph

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