Existence of minimal models for generalized log-canonical pairs
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For projective generalized log-canonical Q-pairs over an algebraically closed characteristic-zero field with a fixed nef rational b-divisor, does a minimal model exist when the adjoint is pseudo-effective, and a Mori fibre space otherwise? The target extracts no divisors and preserves discrepancy inequalities; the Mori fibre space has relative Picard number one.
Why it matters and what counts as progress
Why it matters: The minimal-model/Mori-fibre alternative is a central birational existence target for generalized log-canonical pairs in arbitrary dimension.
What counts as progress: Construct the required birational model in new dimensions or establish the full rational-pair existence theorem under the stated nef-part convention.
How the problem developed
- 4 Mar 2026 · Partial result [1]
Authors of Existence of the minimal model program for log canonical generalized pairs Proved existence of flips for log-canonical generalized pairs without assuming klt, NQC or Q-factoriality, advancing the minimal-model program without proving the full minimal-model existence statement.
- 24 Sept 2026 · Reported result [2]
OpenAI Claimed that projective generalized log-canonical Q-pairs over algebraically closed characteristic-zero fields admit a minimal model when the adjoint divisor is pseudoeffective and a Mori fibre space otherwise, keeping the nef b-divisor fixed.
AI contribution
Minimal models and Mori fibre spaces for generalized log canonical Q-pairs
For projective generalized log-canonical Q-pairs over an algebraically closed characteristic-zero field with a fixed nef rational b-divisor, the manuscript claims a birational model that extracts no divisors and preserves discrepancy inequalities, with nef adjoint when pseudo-effective, or a relative-Picard-number-one Mori fibre space otherwise.
Sources
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