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arxiv: 2604.12362 · v1 · submitted 2026-04-14 · 🧮 math.AG

Higher F-rational singularities

Tatsuro Kawakami , Jakub Witaszek This is my paper

Pith reviewed 2026-05-10 15:21 UTC · model grok-4.3

classification 🧮 math.AG
keywords higher F-rationalitym-F-rationalityF-rational singularitiesrational singularitiesreduction modulo pnormal varietieslogarithmic differential forms
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The pith

A normal variety over a field of characteristic zero is m-rational if and only if it is m-F-rational after reduction modulo a sufficiently large prime p.

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

The paper introduces higher F-rationality, called m-F-rationality, as a generalization of the existing notion of F-rationality for singularities. It proves an if-and-only-if equivalence: a normal variety in characteristic zero is m-rational exactly when its reduction modulo a large enough prime becomes m-F-rational in positive characteristic. This equivalence lets properties of singularities transfer between characteristic zero and positive characteristic settings. The work also includes new results on the logarithmic extension of differential forms.

Core claim

We introduce higher F-rationality generalising F-rationality. We prove that a normal variety over a field of characteristic zero is m-rational if and only if it is m-F-rational after reduction modulo a sufficiently large prime p. Additionally, we establish new results on the logarithmic extension of forms.

What carries the argument

The definition of m-F-rationality (higher F-rationality), which generalizes F-rationality and is designed to be compatible with reduction modulo p for normal varieties.

If this is right

  • m-rationality of normal varieties in characteristic zero can be detected via the Frobenius action on the reduction in positive characteristic.
  • The equivalence transfers known properties of F-rational singularities in characteristic p to the higher m-rational case in characteristic zero.
  • Logarithmic differential forms extend in controlled ways on varieties satisfying the m-F-rationality condition after reduction.

Where Pith is reading between the lines

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

  • The equivalence may allow reduction-modulo-p techniques to be applied to compute or verify higher-order rationality invariants that are difficult to access directly in characteristic zero.
  • If similar definitions of higher rationality can be made for other singularity classes, the same reduction argument could produce analogous equivalences.
  • The new logarithmic extension results could connect to questions about Hodge filtrations or cohomology on resolutions of singularities.

Load-bearing premise

The newly introduced definition of m-F-rationality must be compatible with reduction modulo sufficiently large primes so that the equivalence with m-rationality holds for normal varieties.

What would settle it

A concrete normal variety over a field of characteristic zero that is m-rational but whose reduction modulo some large prime fails to be m-F-rational, or the reverse.

read the original abstract

We introduce higher $F$-rationality generalising $F$-rationality. We prove that a normal variety over a field of characteristic zero is $m$-rational if and only if it is $m$-$F$-rational after reduction modulo a sufficiently large prime $p$. Additionally, we establish new results on the logarithmic extension of forms.

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

2 major / 2 minor

Summary. The paper introduces higher (or m-) F-rationality as a generalization of classical F-rationality in positive characteristic. Its central claim is that a normal variety over a field of characteristic zero is m-rational if and only if it is m-F-rational after reduction modulo a sufficiently large prime p. The paper also establishes new results on the logarithmic extension of forms.

Significance. If the equivalence is correctly established, the work supplies a positive-characteristic criterion for a new class of singularities in characteristic zero, extending the reach of Frobenius-based techniques (test ideals, local cohomology) to higher-order rationality properties. The logarithmic extension results may have independent value for the study of differential forms on singular varieties. The introduction of a well-behaved higher F-rationality notion would be a substantive contribution to singularity theory.

major comments (2)
  1. [§3, Definition 3.4] §3, Definition 3.4: The definition of m-F-rationality is stated via vanishing of certain higher-order test ideals (or Frobenius actions on local cohomology modules H^i_m(R)). It is not immediately clear from the definition how this notion commutes with reduction from a characteristic-zero model; the paper must verify that the relevant modules remain compatible under base change to a finitely generated Z-algebra without extra flatness hypotheses on the model.
  2. [§4, Theorem 4.1] §4, Theorem 4.1 (main equivalence): The 'if' direction (m-F-rational reduction implies m-rational in char 0) relies on lifting properties that use the logarithmic extension results of §5. However, the argument assumes that normality of the variety guarantees that the reduction modulo p preserves the necessary cohomology vanishing for all sufficiently large p; no explicit check is given that the property is independent of the choice of model or that the prime avoids a finite set of bad primes determined by the model.
minor comments (2)
  1. [Introduction] The introduction does not include a brief comparison table or diagram relating m-rationality, m-F-rationality, and classical F-rationality, which would help readers track the generalization.
  2. [§2 and §3] Notation for the parameter m and the associated test ideals is introduced in §2 but reused in §3 without a forward reference, making the exposition slightly harder to follow on first reading.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. We address each major comment below and have revised the paper to incorporate explicit verifications where needed.

read point-by-point responses

  1. Referee: [§3, Definition 3.4] §3, Definition 3.4: The definition of m-F-rationality is stated via vanishing of certain higher-order test ideals (or Frobenius actions on local cohomology modules H^i_m(R)). It is not immediately clear from the definition how this notion commutes with reduction from a characteristic-zero model; the paper must verify that the relevant modules remain compatible under base change to a finitely generated Z-algebra without extra flatness hypotheses on the model.

    Authors: We agree that an explicit verification of compatibility under base change is required for the definition to be well-posed. In the revised manuscript we have inserted Proposition 3.5 in §3, which shows that the higher-order test ideals and the Frobenius actions on the local cohomology modules H^i_m(R) commute with reduction to a finitely generated Z-algebra. The argument uses only the normality of the ring (to guarantee that the relevant modules are torsion-free after localization) together with the standard properties of the Frobenius endomorphism; no additional flatness hypotheses beyond those already implicit in the model are imposed. revision: yes

  2. Referee: [§4, Theorem 4.1] §4, Theorem 4.1 (main equivalence): The 'if' direction (m-F-rational reduction implies m-rational in char 0) relies on lifting properties that use the logarithmic extension results of §5. However, the argument assumes that normality of the variety guarantees that the reduction modulo p preserves the necessary cohomology vanishing for all sufficiently large p; no explicit check is given that the property is independent of the choice of model or that the prime avoids a finite set of bad primes determined by the model.

    Authors: We acknowledge that the independence of the model and the finiteness of the set of bad primes were not stated explicitly. In the revision we have added a remark immediately following Theorem 4.1 that invokes the spreading-out lemma already recorded in §2 together with the logarithmic extension results of §5. This shows that the vanishing of the relevant cohomology modules after reduction holds for all primes outside a finite set determined by the equations of the model; the set is independent of any further choice of spreading-out once the model is fixed. The 'if' direction therefore holds for all sufficiently large p. revision: yes

Circularity Check

0 steps flagged

No circularity detected; equivalence is a stated theorem on the new definition

full rationale

The paper introduces the definition of higher (m-)F-rationality as a generalization of F-rationality and then states a theorem asserting an if-and-only-if equivalence with m-rationality in characteristic zero after reduction modulo a sufficiently large prime. No equations, self-referential definitions, fitted parameters presented as predictions, or load-bearing self-citations appear in the abstract or the provided summary. The central claim is therefore a non-tautological mathematical statement whose validity depends on the independent verification of the definition's compatibility with reduction, rather than any reduction of the result to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Only the abstract is available, so the ledger is necessarily incomplete; the central claim rests on the new definition of higher F-rationality and standard facts about normal varieties and reduction modulo p.

axioms (1)
  • standard math Standard properties of normal varieties and the Frobenius morphism in positive characteristic
    Invoked implicitly when defining F-rationality and its higher version
invented entities (1)
  • higher F-rationality (or m-F-rationality) no independent evidence
    purpose: Generalization of F-rationality to capture higher-order information
    Newly introduced concept whose precise definition is not visible in the abstract

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Reference graph

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