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Math-2351 · mathematics · Algebraic geometry

Weak Zariski Decompositions of Pseudo-Effective Divisors

Status Partially solved · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
Let $D$ be a pseudo-effective relative real Cartier divisor on a normal variety $X/Z$. Must there exist a projective birational map $f:W\to X$ and a numerical decomposition $f^*D\equiv_Z P+N$ with $P$ nef over $Z$ and $N$ effective? Such a decomposition is called a weak Zariski decomposition.

Formulation references: [S1], [S2].

Short English statement

Let $D$ be a pseudo-effective relative real Cartier divisor on a normal variety $X/Z$. Must there exist a projective birational map $f:W\to X$ and a numerical decomposition $f^*D\equiv_Z P+N$ with $P$ nef over $Z$ and $N$ effective? Such a decomposition is called a weak Zariski decomposition.

Sources

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

Math-2351-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2351

Manuscript 1 (family 036, 2026-09-24): A generalized minimal model gives a nef positive part and effective exceptional negative part for the rational adjoints satisfying this manuscript's data assumptions. The problem weak-decomposition question concerns any pseudo-effective real Cartier divisor, not merely these adjoints.

Manuscript 2 (family 036, 2026-10-03): For the specified projective rational klt nef adjoints, the numerical semiample representative is already nef, hence gives a weak numerical Zariski decomposition with zero negative part. Arbitrary pseudo-effective real divisors on arbitrary normal projective morphisms are absent.

Weak Zariski Decompositions of Pseudo-Effective Divisors

OpenAI's claim

Manuscript 1 (family 036, 2026-09-24): Projective generalized log-canonical rational pairs are claimed to admit minimal models in the pseudo-effective case and Mori fiber spaces otherwise, with the chosen nef b-divisor retained. Manuscript 2 (family 036, 2026-10-03): For projective characteristic-zero klt pairs, a nef ordinary adjoint plus a nef rational divisor is claimed to be numerically equivalent to a semiample divisor whenever the ordinary adjoint is pseudo-effective.

Mathematical statement of the claimed result

Mathematical claim 1.
Over a field that has characteristic zero and is algebraically closed, let $(X,B+M)$ be a projective generalized log canonical rational pair: $B\ge0$ is rational, and $M$ is the trace of a rational Cartier divisor, nef on a higher birational model projective over the base, with $D=K_X+B+M$ rational Cartier. OpenAI claims a birational map $\phi:X\dashrightarrow Y$ extracting no divisor, preserving the nef b-divisor, such that $(Y,B_Y+M_Y)$ remains generalized log canonical and the discrepancy comparison is \[ A_{X,B+M}(E)\le A_{Y,B_Y+M_Y}(E) \] for every prime divisor over the function field, strictly for original contracted primes. If $D$ is pseudo-effective, the endpoint adjoint is nef. Otherwise it claims a connected-fiber projective contraction $Y\to T$ with $\dim T<\dim Y$, $\rho(Y/T)=1$ and $-(K_Y+B_Y+M_Y)$ relatively ample. The inequalities use log discrepancies, equal to $1-b$ for a boundary component of coefficient $b$.
Mathematical claim 2.
For a rational projective pair with klt singularities $(X,B)$ over a characteristic-zero field assumed algebraically closed, assume $K_X+B$ pseudo-effective and $M$ nef rational Cartier. If $D=K_X+B+M$ is nef, OpenAI claims \[ \exists\text{ semiample rational Cartier }L\text{ on }X:\quad D\equiv L. \] Numerical equivalence is equality of degrees on every integral curve. Semiampleness is claimed for a numerically equivalent replacement, not necessarily for the original actual divisor $D$.

Outline of the argument

Manuscript 1 (family 036, 2026-09-24): The argument extends positive-boundary programs by selecting valuations with controlled normalized volume and character congruences. Equivariant regrading and finitely generated Rees constructions combine with complements and polarized boundedness to obtain the endpoint while keeping the generalized nef data fixed.

Manuscript 2 (family 036, 2026-10-03): Ordinary log abundance and terminating programs supply the birational reductions. The remaining full-nef-dimension Calabi--Yau case is treated with jet and Frobenius estimates, after which the positive part descends as a semiample numerical representative; flat nontorsion bundles explain why numerical equivalence is the conclusion.

Formal verification

Family 036 has no Lean scope page in this snapshot; no formalization of these manuscript results is listed.

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Status caveat

OpenAI attributes these results to an internal model. Its release notice cautions that results lacking full formalization can contain mistakes. This site has not independently assessed the proofs; a solved label records the scope of the reported claim.

Sources

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