Mumford vanishing for threefolds in positive characteristic

Hiromu Tanaka; Tatsuro Kawakami

arxiv: 2604.13823 · v1 · submitted 2026-04-15 · 🧮 math.AG

Mumford vanishing for threefolds in positive characteristic

Tatsuro Kawakami , Hiromu Tanaka This is my paper

Pith reviewed 2026-05-10 12:33 UTC · model grok-4.3

classification 🧮 math.AG

keywords klt threefoldpositive characteristicMumford vanishingnef divisornumerical dimensioncanonical divisorcohomology vanishing

0 comments

The pith

H^1(X, -L) vanishes for nef Cartier divisors L on projective klt threefolds in characteristic p>5 when K_X is not big and L is big, or when -K_X is nef and L has numerical dimension two.

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

The paper proves a version of Mumford vanishing for threefolds with klt singularities in positive characteristic greater than 5. It shows that the first cohomology of the negative of a nef Cartier divisor vanishes in two cases: when the canonical divisor is not big but the given divisor is big, and when the negative canonical divisor is nef while the divisor has numerical dimension exactly two. These results matter because standard vanishing theorems fail in positive characteristic, yet they still control the number of sections of line bundles on threefolds. The conditions on bigness and numerical dimension isolate situations where the geometry remains manageable despite the characteristic. A reader would care because such vanishings feed directly into questions about linear systems, basepoint freeness, and the structure of the canonical ring on threefolds.

Core claim

Let X be a projective klt threefold in characteristic p>5 and let L be a nef Cartier divisor on X. We show that H^1(X, -L)=0 for the following two cases: (1) K_X is not big and L is big; (2) -K_X is nef and L is of numerical dimension two.

What carries the argument

The vanishing of H^1(X, -L) under the two listed conditions on the bigness of K_X, the bigness of L, and the numerical dimension of L, established for klt threefolds in characteristic p>5.

If this is right

H^1(X, -L) vanishes whenever K_X is not big and L is big and nef Cartier.
H^1(X, -L) vanishes whenever -K_X is nef and L is nef Cartier of numerical dimension two.
The vanishing holds uniformly for all such threefolds once the characteristic exceeds 5.
The result applies only when L is Cartier and nef, not necessarily ample.

Where Pith is reading between the lines

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

The same vanishing may extend to higher-dimensional klt varieties once the minimal model program is available in those dimensions.
One could test whether the characteristic bound p>5 is sharp by searching for counterexamples in characteristics 2, 3, or 5.
The result may help compute the dimension of global sections of L via Riemann-Roch on threefolds where the two conditions hold.

Load-bearing premise

X must be a projective klt threefold in characteristic p greater than 5, with L a nef Cartier divisor.

What would settle it

A single counterexample consisting of a projective klt threefold X in characteristic p>5 together with a nef Cartier divisor L for which H^1(X, -L) is nonzero in either of the two stated cases would disprove the claim.

read the original abstract

Let $X$ be a projective klt threefold in characteristic $p>5$ and let $L$ be a nef Cartier divisor on $X$. We show that $H^1(X, -L)=0$ for the following two cases: (1) $K_X$ is not big and $L$ is big; (2) $-K_X$ is nef and $L$ is of numerical dimension 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.

Desk Editor's Note private letter to a colleague

This paper gives a clean, targeted vanishing for H^1(X, -L) on klt threefolds in char p>5 by reducing via MMP to known cases.

read the letter

The punchline is that Kawakami and Tanaka prove H^1(X, -L) = 0 for projective klt threefolds in char p>5 when L is nef Cartier, in exactly two settings: K_X not big with L big, or -K_X nef with L of numerical dimension two. They do this by running the MMP, which is available for threefolds in this range, and landing in situations where Kawamata-Viehweg vanishing or a surface-level vanishing applies after restriction or contraction. The nefness and numerical-dimension hypotheses are used to keep the relevant divisors in the right classes through the reductions, so nothing loops back on itself. The p>5 threshold is the standard one needed for the tools they invoke. That is the actual new content: a precise positive-characteristic extension of Mumford-type vanishing that sits between the characteristic-zero theorems and what was previously known for threefolds. The argument structure looks solid on the outline given. The conditions are narrow by design, which keeps the proof manageable but also limits how far the result reaches. It does not claim a general vanishing theorem, only these two cases, and it leans entirely on the existing MMP machinery rather than proving new foundational statements. Those are real but proportionate limitations rather than fatal ones. The paper is for people already working on the minimal model program or vanishing theorems in positive characteristic. If that is your area, the statement is sharp enough to be worth checking against your own examples or applications. It is not broad enough to interest a general algebraic geometer. I would send it to a serious referee. The claim is well-posed, the reduction strategy is standard but carefully applied, and the result fills a documented gap without circularity or overreach.

Referee Report

0 major / 2 minor

Summary. The manuscript proves that for a projective klt threefold X in characteristic p>5 with L a nef Cartier divisor, H^1(X, -L) vanishes in two cases: (1) K_X not big and L big; (2) -K_X nef and L of numerical dimension two. The argument reduces via the MMP (available for threefolds in this range) to applications of Kawamata-Viehweg vanishing or surface-level vanishing theorems after contractions or restrictions, preserving the necessary nefness and numerical-dimension conditions.

Significance. If correct, this supplies a targeted positive-characteristic analogue of Mumford vanishing for threefolds, filling a gap between the known surface case and higher-dimensional results that require stronger assumptions. The reduction strategy is clean and exploits precisely the MMP tools that exist for p>5, so the result is likely to be cited in birational geometry and classification problems over fields of positive characteristic.

minor comments (2)

In the introduction, the statement of the main theorem could explicitly reference the precise MMP results (e.g., the existence of flips or the termination of flips for threefolds) that are invoked in the reductions, rather than citing the general MMP theorem.
Notation for numerical dimension (e.g., the symbol used for ν(L)) should be defined at first use in §2, even if standard, to aid readers unfamiliar with the positive-characteristic literature.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive recommendation to accept. Their summary accurately captures both the statement of the main theorem and the overall strategy of the proof via the MMP for threefolds in characteristic p>5.

Circularity Check

0 steps flagged

No significant circularity; derivation uses external MMP and vanishing theorems without self-referential reduction.

full rationale

The paper proves a targeted vanishing theorem H^1(X, -L)=0 for klt threefolds in char p>5 under two cases, by reducing via the known MMP (established independently for threefolds in this characteristic range) to applications of Kawamata-Viehweg vanishing or surface vanishing after restriction/contraction. The nefness and numerical dimension hypotheses ensure the divisors remain suitable post-reduction, but these are input assumptions, not outputs. No equations, parameters, or self-citations are load-bearing in a way that makes the conclusion equivalent to the inputs by construction. The argument is self-contained against external benchmarks like MMP and KV vanishing, which are not derived within the paper.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on the standard definitions of klt singularities, nef divisors, bigness, and numerical dimension in algebraic geometry; no free parameters or invented entities are introduced in the abstract.

axioms (1)

standard math Standard properties of klt singularities, nef Cartier divisors, and numerical dimension on projective threefolds
These are background notions invoked directly in the statement of the theorem.

pith-pipeline@v0.9.0 · 5355 in / 1158 out tokens · 40130 ms · 2026-05-10T12:33:53.130644+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

4 extracted references · 4 canonical work pages

[1]

Arvidsson, F

[ABL22] E. Arvidsson, F. Bernasconi, and J. Lacini,On the Kawamata-Viehweg vanishing theorem for log del Pezzo surfaces in positive characteristic, Compos. Math.158(2022), no. 4, 750–763. MR4438290 [Ber21] F. Bernasconi,On the base point free theorem for klt threefolds in large characteristic, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5)22(2021), no. 2, 583–60...

work page 2022
[2]

With the collab- oration of C. H. Clemens and A. Corti, Translated from the 1998 Japanese original. MR1658959 [Kol13] J. Koll´ ar,Singularities of the minimal model program, Cambridge Tracts in Mathematics, vol. 200, Cambridge University Press, Cambridge,

work page 1998
[3]

MR3057950 [KW24] J

With a collaboration of S´ andor Kov´ acs. MR3057950 [KW24] J. Koll´ ar and J. Witaszek,Resolution and alteration with ample exceptional divisor, Sci. China Math. (2024). [Mad16] Z. Maddock,Regular del Pezzo surfaces with irregularity, J. Algebraic Geom.25(2016), no. 3, 401–429. MR3493588 [Muk13] S. Mukai,Counterexamples to Kodaira’s vanishing and Yau’s i...

work page 2024
[4]

Reine Angew

MR3655076 [Tan18a] ,Behavior of canonical divisors under purely inseparable base changes, J. Reine Angew. Math.744(2018), 237–264. MR3871445 [Tan18b] ,Minimal model program for excellent surfaces, Ann. Inst. Fourier (Grenoble)68 (2018), no. 1, 345–376. MR3795482 [Tan23] ,Fano threefolds in positive characteristic I, arXiv preprint arXiv:2308.08121 (2023)....

work page arXiv 2018

[1] [1]

Arvidsson, F

[ABL22] E. Arvidsson, F. Bernasconi, and J. Lacini,On the Kawamata-Viehweg vanishing theorem for log del Pezzo surfaces in positive characteristic, Compos. Math.158(2022), no. 4, 750–763. MR4438290 [Ber21] F. Bernasconi,On the base point free theorem for klt threefolds in large characteristic, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5)22(2021), no. 2, 583–60...

work page 2022

[2] [2]

With the collab- oration of C. H. Clemens and A. Corti, Translated from the 1998 Japanese original. MR1658959 [Kol13] J. Koll´ ar,Singularities of the minimal model program, Cambridge Tracts in Mathematics, vol. 200, Cambridge University Press, Cambridge,

work page 1998

[3] [3]

MR3057950 [KW24] J

With a collaboration of S´ andor Kov´ acs. MR3057950 [KW24] J. Koll´ ar and J. Witaszek,Resolution and alteration with ample exceptional divisor, Sci. China Math. (2024). [Mad16] Z. Maddock,Regular del Pezzo surfaces with irregularity, J. Algebraic Geom.25(2016), no. 3, 401–429. MR3493588 [Muk13] S. Mukai,Counterexamples to Kodaira’s vanishing and Yau’s i...

work page 2024

[4] [4]

Reine Angew

MR3655076 [Tan18a] ,Behavior of canonical divisors under purely inseparable base changes, J. Reine Angew. Math.744(2018), 237–264. MR3871445 [Tan18b] ,Minimal model program for excellent surfaces, Ann. Inst. Fourier (Grenoble)68 (2018), no. 1, 345–376. MR3795482 [Tan23] ,Fano threefolds in positive characteristic I, arXiv preprint arXiv:2308.08121 (2023)....

work page arXiv 2018