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Math-2548 · mathematics · Algebraic geometry

Minimal or Mori models for generalized rational lc pairs

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Let $k$ be an algebraically closed field of characteristic zero. Let $(X,B+M)$ be a normal integral projective generalized log canonical rational pair over $k$, with its nef rational b-Cartier data $\mathbf M$ fixed and $M$ the trace on $X$. Put $D=K_X+B+M$.

Must there exist a normal integral projective variety $Y$ and a birational map $\phi:X\dashrightarrow Y$, whose inverse contracts no divisors, such that $B_Y=\phi_*B$, $M_Y$ is the trace of the same $\mathbf M$, and \[ A_{X,B+M}(E)\le A_{Y,B_Y+M_Y}(E) \] for every prime divisor $E$ over the common function field, with strict inequality for each prime divisor on $X$ contracted by $\phi$? Here $A$ denotes generalized log discrepancy.

The required model must have $K_Y+B_Y+M_Y$ nef when $D$ is pseudo-effective. Otherwise it must admit a projective contraction $h:Y\to T$ with connected fibres to a lower-dimensional normal projective base, satisfying \[ \rho(Y/T)=1, \qquad -(K_Y+B_Y+M_Y)\text{ is ample over }T. \] The nef data may not be replaced by a different b-divisor.

Short English statement

Over an algebraically closed field of characteristic zero, does every normal integral projective generalized log canonical rational pair $(X,B+M)$, with fixed nef rational b-Cartier data, have a projective birational model $Y$ extracting no divisor, on which generalized discrepancies do not decrease and increase strictly for contracted prime divisors on $X$? The adjoint on $Y$ must be nef if the original adjoint is pseudo-effective. Otherwise $Y$ must have a contraction with connected fibres to a lower-dimensional normal projective base, relative Picard number one, and relatively antiample adjoint. The traces on both models must come from the same fixed nef b-divisor.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2548-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2548

The source ranges over projective generalized log canonical rational pairs over a characteristic-zero field assumed algebraically closed, with fixed nef rational b-Cartier data. Its model extracts no divisor, has nondecreasing generalized log discrepancies with strict increase on contracted original primes, and gives the nef or relative-Picard-one Mori alternative according to pseudo-effectivity. The source field convention and rational-pair restrictions are part of this formulation.

Minimal or Mori models for generalized rational lc pairs

OpenAI's claim

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. 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, 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 $b$.

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.

Formal verification

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

This site has not executed the Lean files or checked their dependencies. These descriptions report the scope of the linked declarations, rather than a new verification of the proofs.

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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