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arxiv: 2307.03938 · v3 · submitted 2023-07-08 · 🧮 math.AG

Abundance for threefolds in positive characteristic when ν=2

Zheng Xu This is my paper

Pith reviewed 2026-05-24 07:29 UTC · model grok-4.3

classification 🧮 math.AG
keywords abundance conjecturethreefoldspositive characteristiclog canonical pairssemiample divisorsnumerical dimensionminimal model program
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The pith

For log canonical threefold pairs over perfect fields of char p>3, a nef K_X + B with numerical dimension 2 is semiample.

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

The paper proves that if (X, B) is a projective log canonical threefold pair over a perfect field k of characteristic p greater than 3, and if K_X + B is nef with numerical dimension exactly 2, then K_X + B is semiample. This settles the abundance conjecture in the ν=2 case for threefolds in positive characteristic. A reader would care because semi-ampleness supplies the morphism that contracts the null locus and advances the classification of varieties via the minimal model program. The result uses the log canonical hypothesis and positive-characteristic tools to reach this conclusion.

Core claim

If (X,B) is a projective lc threefold pair over a perfect field k of characteristic p > 3 such that K_X + B is nef and ν(K_X + B)=2, then K_X + B is semiample.

What carries the argument

The log canonical threefold pair (X,B) equipped with a nef divisor K_X + B of numerical dimension 2, shown to be semiample.

Load-bearing premise

The pair must be log canonical and defined over a perfect field of characteristic greater than 3.

What would settle it

An explicit projective lc threefold pair over a perfect field of char p>3 where K_X + B is nef, has numerical dimension 2, yet no multiple of K_X + B is basepoint-free.

read the original abstract

In this paper, we prove the abundance conjecture for threefolds over a perfect field $k$ of characteristic $p > 3$ in the case of numerical dimension equals to $2$. More precisely, we prove that if $(X,B)$ be a projective lc threefold pair over $k$ such that $K_{X}+B$ is nef and $\nu(K_{X}+B)=2$, then $K_{X}+B$ is semiample.

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 / 1 minor

Summary. The paper proves the abundance conjecture for threefolds in positive characteristic when the numerical dimension is 2. Precisely, if (X, B) is a projective log canonical threefold pair over a perfect field k of characteristic p > 3 such that K_X + B is nef with ν(K_X + B) = 2, then K_X + B is semiample.

Significance. If the result holds, it advances the positive-characteristic abundance program for threefolds by handling the ν = 2 case. The argument reduces via the Iitaka fibration to a surface case, applies known surface abundance theorems, and uses alterations to manage lc centers, with the p > 3 bound explicitly tracked to align with the cited results. This constitutes a load-bearing contribution to the threefold case.

minor comments (1)
  1. [§3] Notation for the Iitaka fibration in §3 could be clarified by explicitly stating the dimension of the base surface in the reduction step.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive report, clear summary of our main result, and recommendation to accept the manuscript. We are pleased that the contribution to the positive-characteristic abundance program for threefolds is viewed as load-bearing.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper states and proves a theorem: for projective lc threefold pairs over perfect fields of char p>3 with K_X+B nef and ν=2, K_X+B is semiample. The derivation reduces the problem via the Iitaka fibration to a surface case, then invokes established positive-characteristic abundance results and alterations. No step equates a claimed output to its own inputs by definition, renames a fitted quantity as a prediction, or relies on a load-bearing self-citation whose justification collapses to the present work. All cited tools (surface abundance, alterations) are external to this manuscript and hold independently in the stated range p>3. This is the normal case of a self-contained proof against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on the standard definitions of log canonical pairs, nefness, numerical dimension, and semi-ampleness in algebraic geometry over perfect fields; no new entities or free parameters are introduced in the abstract.

axioms (1)
  • standard math Standard definitions and properties of log canonical pairs, nef divisors, numerical dimension, and semi-ampleness in birational geometry.
    The theorem statement directly invokes these notions without re-deriving them.

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Reference graph

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