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Math-2591 · mathematics · Algebraic geometry

Uniform Cartier-section thresholds for Fano-type contractions

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For every $d>0$ and real $\varepsilon >0$, does $\tau (d,\varepsilon )>0$ exist such that each $\varepsilon -lc$ rational d-dimensional pair (X,B), Fano-type contraction $f:X\to Z$ of normal integral quasi-projective varieties with dim $Z>0$ and relatively nef $-(K_X+B)$, and closed $z\in Z$ admits a neighborhood U and nonzero effective Cartier D on U through z with $(f^-1(U),B+\tau $ f*D) lc? Fields are algebraically closed of characteristic zero.

Short English statement

For every $d>0$ and real $\varepsilon >0$, does $\tau (d,\varepsilon )>0$ exist such that each $\varepsilon -lc$ rational d-dimensional pair (X,B), Fano-type contraction $f:X\to Z$ of normal integral quasi-projective varieties with dim $Z>0$ and relatively nef $-(K_X+B)$, and closed $z\in Z$ admits a neighborhood U and nonzero effective Cartier D on U through z with $(f^-1(U),B+\tau $ f*D) lc? Fields are algebraically closed of characteristic zero.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2591-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2591

Manuscript 1 (family 066, 2026-09-25): The problem selects the mathematical statement summarized here, including its stated hypotheses and parameter range. The cited theorem gives a claimed proof of that assertion in full.

Uniform Cartier-section thresholds for Fano-type contractions

OpenAI's claim

For the stated Fano-type contractions and any chosen base point, a Cartier divisor containing that point is claimed to have a pullback threshold bounded below uniformly in dimension and discrepancy gap. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
Over a field that has characteristic zero and is algebraically closed, fix $d>0$ and real $\varepsilon>0$. Let $(X,B)$ be an effective rational pair of dimension $d$ with all log discrepancies at least $\varepsilon$ and $-(K_X+B)$ nef over a Fano-type contraction $f:X\to Z$, with $X$ and $Z$ normal, integral and quasi-projective, $\dim Z>0$. Fano type means that some effective rational auxiliary $C$ makes $(X,C)$ klt and $-(K_X+C)$ relatively ample. OpenAI claims a uniform $\tau(d,\varepsilon)>0$ such that for every closed $z\in Z$ there is a neighborhood $U$ and nonzero effective Cartier $D$ on $U$ through $z$ with \[ (f^{-1}(U),B|_{f^{-1}(U)}+\tau f^*D)\text{ log canonical}. \] The threshold controls the whole inverse image of the neighborhood, independent of indices and boundary denominators; $D$ and $U$ may depend on $z$ and the contraction.

Outline of the argument

Manuscript 1 (family 066, 2026-09-25): Ratios of opposite character orders impose exact invariant relations on a constrained valuation slice. Finitely generated central algebras retain another independent variable unless they produce a quotient prime whose discrepancy tends to zero. Finite ambient dimension prevents unlimited variable growth, while the positive discrepancy gap excludes the alternative.

Formal verification

Family 066 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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