---
title: "Log-canonical threefold abundance in numerical dimension one"
authors:
  - "OpenAI (internal model)"
year: 2026
venue: "openai/math manuscript collection (GitHub), result family 035; unreviewed"
arxiv: null
doi: null
source: "https://github.com/openai/math/blob/main/preprints/Log-abundance-in-numerical-dimension-one-for-threefolds-in-positive-characteristic-October-5-2026/paper.pdf"
topics:
  - "openai-math"
  - "ai-generated-math"
  - "algebraic-and-complex-geometry"
  - "unformalized"
seed_rank: 1895
seed_batch: "openai-math-2026-10-07"
reviewed: "2026-10-07"
pool: "maths-foundations"
relevance_score: 7
lineage: ai-mathematical-reasoning
cites:
  - title: "Log abundance in numerical dimension one for threefolds in positive characteristic"
    url: "https://github.com/openai/math/blob/main/preprints/Log-abundance-in-numerical-dimension-one-for-threefolds-in-positive-characteristic-October-5-2026/paper.pdf"
    year: 2026
    arxiv: null
    doi: null
  - title: "Abundance in numerical dimension one for terminal threefolds in positive characteristic"
    url: "https://github.com/openai/math/blob/main/preprints/Abundance-in-numerical-dimension-one-for-terminal-threefolds-in-positive-characteristic-September-24-2026/paper.pdf"
    year: 2026
    arxiv: null
    doi: null
---

# Log-canonical threefold abundance in numerical dimension one

## One-sentence takeaway

OpenAI's result family 035 (Algebraic and complex geometry) claims: Proves log abundance for projective log canonical threefold pairs over algebraically closed fields of characteristic p > 3 when the effective boundary is rational and the ℚ-Cartier adjoint is nef of numerical dimension one.

## Why it matters here

Pure mathematics with no direct engine use; it is in the library as part of the openai/math corpus, which records what an AI research model claims to have proved. More broadly, every card in this batch is evidence for the `ai-mathematical-reasoning` thread: per the repo README, an unreleased internal OpenAI model produced it, using about three hours of ChatGPT Pro thinking compute per result on average, across roughly 4,000 posed problems.

## Key ideas

- Family summary (repo): The adjoint is semiample, without requiring the original variety to be terminal or ℚ-factorial.
- *Log abundance in numerical dimension one for threefolds in positive characteristic*: We prove the numerical-dimension-one case of log abundance for threefolds over algebraically closed fields of characteristic p > 3. If $(X,B)$ is a projective log canonical threefold pair with effective rational boundary, and $K_X+B$ is ℚ-Cartier, nef, and of numerical dimension one, then $K_X+B$ is semiample. Neither terminality nor ℚ-factoriality of X is required.
- *Abundance in numerical dimension one for terminal threefolds in positive characteristic*: 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 $\nu(K_X)=1$, then KX is semiample and $\kappa(X,K_X)=1$.

## Caveats

- Unreviewed: these are AI-generated manuscripts posted to GitHub with no peer review or arXiv deposit. The repo README says results are "at different stages of verification" and that "some of the unformalized results could have issues."
- This card restates the repo's family summary, manuscript abstracts and Lean scope notes. It is not an independent check of the proofs, and the scope of each theorem is exactly as stated there.
- Lean: not formalized. The repo has no Lean development for this family, so the claim rests on the prose manuscript(s) alone.

## Links

- Manuscript: [Log abundance in numerical dimension one for threefolds in positive characteristic](https://github.com/openai/math/blob/main/preprints/Log-abundance-in-numerical-dimension-one-for-threefolds-in-positive-characteristic-October-5-2026/paper.pdf)
- Manuscript: [Abundance in numerical dimension one for terminal threefolds in positive characteristic](https://github.com/openai/math/blob/main/preprints/Abundance-in-numerical-dimension-one-for-terminal-threefolds-in-positive-characteristic-September-24-2026/paper.pdf)
- Family entry: https://github.com/openai/math/blob/main/CONTENTS.md
- Overview PDF: https://github.com/openai/math/blob/main/overview.pdf
- Repo: https://github.com/openai/math