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arxiv: 2009.07158 · v2 · pith:W5KX2HYZnew · submitted 2020-09-15 · 🧮 math.AG

Pseudo-effectivity of the relative canonical divisor and uniruledness in positive characteristic

Zsolt Patakfalvi This is my paper

Pith reviewed 2026-05-24 14:27 UTC · model grok-4.3

classification 🧮 math.AG
keywords pseudo-effectiverelative canonical divisorunirulednesspositive characteristiccyclic coverscohomological criterionalgebraic geometry
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The pith

If the generic fiber is non-uniruled, the relative canonical divisor is pseudo-effective in positive 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 for a surjective morphism f from X to T between smooth projective varieties over an algebraically closed field of positive characteristic, if the generic fiber is geometrically integral and non-uniruled, then the relative canonical divisor K_{X/T} is pseudo-effective. This follows from a covering argument: assuming the contrary allows X to be covered by rational curves, which forces a contradiction unless the base admits a suitable non-uniruled cover. The authors construct such a cover as a cyclic cover of degree d not divisible by p, obtained from a general section of a high power of an ample line bundle on T. They establish this by proving a cohomological criterion that a smooth projective variety of dimension n is not uniruled when the dimension of the semi-stable part of its top cohomology with the structure sheaf exceeds that of the previous degree. Singular versions of the statements are proved as well.

Core claim

If f∶X→T is a surjective morphism between smooth projective varieties over an algebraically closed field k of characteristic p>0 with geometrically integral and non-uniruled generic fiber, then K_{X/T} is pseudo-effective. The proof shows existence of a finite smooth non-uniruled cover of T via cyclic covers and uses the cohomological condition that T of dimension n is not uniruled whenever the semi-stable part of H^n(T, O_T) has dimension larger than H^{n-1}(T, O_T).

What carries the argument

The cyclic cover of the base T of degree d not divisible by p, given by a general section of A^d for ample enough A, which produces a non-uniruled cover yielding the contradiction with rational curve coverings.

If this is right

  • The statement extends to singular versions of the varieties involved.
  • The cohomological condition gives an independent test for non-uniruledness based on Hodge numbers with the structure sheaf.
  • Pseudo-effectivity of the relative canonical follows whenever the generic fiber satisfies the non-uniruled hypothesis.
  • The covering construction applies whenever the base admits sufficiently positive line bundles.

Where Pith is reading between the lines

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

  • The technique of producing non-uniruled covers may apply to other positivity questions for divisors on fibrations.
  • The result constrains the possible geometry of fibrations in positive characteristic when fibers avoid rational curves.

Load-bearing premise

A cyclic cover of degree d with p not dividing d, given by a general section of an ample enough line bundle A^d on T, is not uniruled.

What would settle it

A concrete surjective morphism f from X to T with geometrically integral non-uniruled generic fiber where K_{X/T} fails to be pseudo-effective.

read the original abstract

We show that if $f\colon X \to T$ is a surjective morphism between smooth projective varieties over an algebraically closed field $k$ of characteristic $p>0$ with geometrically integral and non-uniruled generic fiber, then $K_{X/T}$ is pseudo-effective. The proof is based on covering $X$ with rational curves, which gives a contradiction as soon as both the base and the generic fiber are not uniruled. However, we assume only that the generic fiber is not uniruled. Hence, the hardest part of the proof is to show that there is a finite smooth non-uniruled cover of the base for which we show the following: If $T$ is a smooth projective variety over $k$ and $\mathcal{A}$ is an ample enough line bundle, then a cyclic cover of degree $p \nmid d$ given by a general element of $\left|\mathcal{A}^d\right|$ is not uniruled. For this we show the following cohomological uniruledness condition, which might be of independent interest: A smooth projective variety $T$ of dimenion $n$ is not uniruled whenever the dimension of the semi-stable part of $H^n(T, \mathcal{O}_T)$ is greater than that of $H^{n-1}(T, \mathcal{O}_T)$. Additionally, we also show singular versions of all the above statements.

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 / 0 minor

Summary. The paper proves that if f: X → T is a surjective morphism between smooth projective varieties over an algebraically closed field k of characteristic p > 0 with geometrically integral and non-uniruled generic fiber, then K_{X/T} is pseudo-effective. The proof proceeds by contradiction: non-pseudo-effectivity implies a covering family of rational curves on X; since the generic fiber is non-uniruled by assumption, the argument reduces to constructing a finite smooth non-uniruled cover T' → T (via a cyclic cover of degree d with p ∤ d cut out by a general section of A^d for A ample) so that the pulled-back family would uniruled both base and fiber. Non-uniruledness of T' is deduced from a new cohomological criterion: a smooth projective variety of dimension n is not uniruled if dim(semi-stable part of H^n(T, O_T)) > dim H^{n-1}(T, O_T). Singular versions of the statements are also given.

Significance. If correct, the result extends pseudo-effectivity statements for relative canonical divisors to positive characteristic under a non-uniruledness hypothesis on the generic fiber, with potential consequences for the minimal model program in char p. The cohomological criterion is presented as possibly of independent interest and is used to handle the base-cover construction, which is the novel technical step. The paper supplies explicit constructions rather than reducing to prior results.

major comments (2)
  1. [cyclic cover construction and application of cohomological criterion] The central contradiction argument (abstract and proof outline) rests on the claim that a general cyclic cover T' → T of degree d (p ∤ d) satisfies the cohomological non-uniruledness criterion dim(semi-stable part of H^n(T', O_{T'})) > dim H^{n-1}(T', O_{T'}). The manuscript must supply the explicit computation or vanishing argument showing this inequality holds for general such covers; without it the reduction to a non-uniruled base fails.
  2. [cohomological uniruledness condition] The definition and computation of the 'semi-stable part' of H^n(T, O_T) (used in the cohomological criterion) must be stated precisely, including its behavior under the cyclic cover and any dependence on the choice of ample line bundle A; this is load-bearing for applying the criterion to produce the required non-uniruled T'.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for identifying points where additional details would strengthen the exposition. We will revise the manuscript accordingly to address both major comments.

read point-by-point responses

  1. Referee: [cyclic cover construction and application of cohomological criterion] The central contradiction argument (abstract and proof outline) rests on the claim that a general cyclic cover T' → T of degree d (p ∤ d) satisfies the cohomological non-uniruledness criterion dim(semi-stable part of H^n(T', O_{T'})) > dim H^{n-1}(T', O_{T'}). The manuscript must supply the explicit computation or vanishing argument showing this inequality holds for general such covers; without it the reduction to a non-uniruled base fails.

    Authors: We agree that the verification of the inequality dim(semi-stable part of H^n(T', O_{T'})) > dim H^{n-1}(T', O_{T'}) for a general cyclic cover was not presented with full explicit computations in the original text. In the revision we will add a dedicated computation section that tracks the cohomology under the cyclic cover of degree d (p ∤ d) ramified along a general section of A^d. This will show that the semi-stable summand in degree n grows by a factor depending on d while the H^{n-1} term remains unchanged or is controlled by the base, establishing the strict inequality for sufficiently ample A and general choice. revision: yes

  2. Referee: [cohomological uniruledness condition] The definition and computation of the 'semi-stable part' of H^n(T, O_T) (used in the cohomological criterion) must be stated precisely, including its behavior under the cyclic cover and any dependence on the choice of ample line bundle A; this is load-bearing for applying the criterion to produce the required non-uniruled T'.

    Authors: We accept that a precise definition and the functoriality under cyclic covers were insufficiently spelled out. The revised manuscript will contain an expanded subsection that defines the semi-stable part of H^n(T, O_T) as the subspace of slope zero in the Harder-Narasimhan filtration of the de Rham cohomology (or equivalently the part invariant under a suitable power of Frobenius in this setting). We will then compute its pull-back behavior under the cyclic cover and verify that the inequality is independent of the particular sufficiently ample A, depending only on the generality of the branch divisor. revision: yes

Circularity Check

0 steps flagged

No significant circularity; new criterion proved and applied independently

full rationale

The derivation proceeds by contradiction assuming K_{X/T} is not pseudo-effective, producing a covering family of rational curves on X. With the generic fiber assumed non-uniruled, the argument requires a non-uniruled finite cover T' of T, obtained via a cyclic cover of degree d (p not dividing d) from a general section of an ample A^d. Non-uniruledness of T' is established via a new cohomological criterion (dim of semi-stable part of H^n(T',O) > dim H^{n-1}(T',O)) that is proved directly in the paper and flagged as potentially of independent interest. No quoted step reduces the target statement to a fitted parameter, self-definition, or load-bearing self-citation chain; the central claim retains independent content from the explicit construction and criterion.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on standard algebraic geometry axioms about projective varieties, line bundles, and cohomology, plus the explicit assumption that the generic fiber is non-uniruled.

axioms (1)
  • standard math Standard properties of cohomology groups, ample line bundles, and cyclic covers on smooth projective varieties over algebraically closed fields.
    Invoked throughout the statements about uniruledness and pseudo-effectivity.

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