Browse › Algebraic and complex geometry › 036 Numerical semiampleness and generalized minimal models

Minimal models and Mori fibre spaces for generalized log canonical Q-pairs

OpenAI · September 24, 2026

Open PDF Cite Report a problem Add a reference or comment Interested in working on this Start a discussion

Edit NOTES.md · Existing issues · Source on GitHub

Abstract

We resolve the existence form of the minimal-model conjecture for projective generalized log canonical ℚ-pairs over algebraically closed fields of characteristic zero. Such a pair admits a minimal model when its adjoint divisor is pseudo-effective, and a Mori fibre space otherwise, while keeping the nef ℚ-Cartier b-divisor fixed. Taking the nef part to be zero gives the corresponding result for ordinary log canonical ℚ-pairs.

If the PDF does not display here, open it directly: Open PDF.

Notes

No notes yet. Add a reference, a comment or your name with the buttons above, or edit NOTES.md on GitHub.

Automatically collected notes (unreviewed)

work in progress

Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.

Possibly relevant literature (found by the agent, not checked)

C. D. Hacon and J. Moraga, On weak Zariski decompositions and termination of flips (arXiv:1805.01600, 2018–2020). Shows that weak Zariski decompositions give minimal models for pseudo-effective generalized klt pairs, and that flips terminate for generalized lc pairs with weak Zariski decompositions (assuming termination in lower dimensions); this is the same weak-Zariski-decomposition route to existence that Theorem 1.1 takes.
Kenta Hashizume, On the non-vanishing conjecture and existence of log minimal models (arXiv:1609.00121; Publ. RIMS 2018). Reduces the log minimal model conjecture for projective lc pairs to non-vanishing for smooth varieties; this is a reduction from lc pairs to the smooth case, comparable to the smooth-to-lc transfer the paper uses.
V. Lazić and Z. Xie, Nakayama-Zariski decomposition and the termination of flips (arXiv:2305.01752; Épijournal Géom. Algébrique 9, 2025). Shows that termination of one sequence of flips implies termination of all, assuming a conjecture on Nakayama–Zariski decompositions; relevant to the paper's remark that it gives a terminating choice of program, not termination of every MMP.

Digests and expositions

AI Proved What?, Semiample, numerically? Generalized abundance (Episode 036, October 2026). This AI-assisted video and transcript outlines all four manuscripts, emphasizing numerical equivalence and dependencies on other release papers. [AI-generated per source] [family-level]

News and blogs

u/mbintz, Some recurring tricks in the OpenAI math release (r/OpenAI, accessed 2026-10-21). The author reports an AI-assisted comparison of this manuscript’s Lemma 8.3 with Proposition 14.3 of Bounded klt complements for Fano contractions, also identifying broader overlapping passages.

Elsewhere