OpenAI · 035-02 · 2026-09-24 · Algebraic and complex geometry
Abundance in numerical dimension one for terminal threefolds in positive characteristic
preprints/Abundance-in-numerical-dimension-one-for-terminal-threefolds-in-positive-characteristic-September-24-2026
Family 035. Threefold log abundance in numerical dimension one in characteristic .
The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.
Read against the standing problems
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
Proves log abundance for projective log canonical threefold pairs over algebraically closed fields of characteristic when the effective boundary is rational and the -Cartier adjoint is nef of numerical dimension one. The adjoint is semiample, without requiring the original variety to be terminal or -factorial.
Let X be a projective ℚ-factorial terminal threefold over an algebraically closed field of characteristic p > 3. We prove the numerical-dimension-one case of abundance: if KX is nef with , then KX is semiample and .
Preprint source and Lean scope
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