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Log abundance in numerical dimension one for threefolds in positive characteristic
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Abstract
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.
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Comments and discussion
- mahrud and math-commentary-bot, 035. Log-canonical threefold abundance in numerical dimension one (PR #35, mahrud/openai-math-comments) (GitHub, 9 Oct 2026). Community comment thread for this family: the repository owner points to arXiv:2610.10439 as "Noteworthy", quotes its authors' AI-use statement, and labels the family "2nd-proof". [family-level]
Related prior literature mentioned by others
- Zheng Xu, Note on the 3-dimensional log canonical abundance in characteristic $>3$ (arXiv:2212.04039, 2022; revised 2024-02-03). The AI Proved What? exposition identifies this work's nonvanishing and semiampleness results for positive Kodaira dimension or nef dimension at most two as prior inputs to this family's numerical-dimension-one claims. [family-level]
- Zheng Xu, Abundance for threefolds in positive characteristic when $ν=2$ (arXiv:2307.03938, 2023; revised 2026-04-17). The AI Proved What? exposition places this family's numerical-dimension-one claims alongside Xu's result for nef log canonical divisors of numerical dimension two in characteristic $p>3$. [family-level]
Possibly relevant literature (found by the agent, not checked)
- Fabio Bernasconi, Jakub Witaszek, Zheng Xu, Abundance for threefolds in positive characteristic (arXiv:2610.10439, 7 Oct 2026). Independent preprint proving abundance for projective lc threefold pairs in characteristic $p>42$ via a reduction to the case $\nu(K_X+B)=1$; its AI-use statement says the authors posted it "following the release of OpenAI's list of solved problems in mathematics"; it does not appear to cite this manuscript. [family-level]
- Omprokash Das and Joe Waldron, On the Abundance Problem for 3-folds in characteristic p>5 (arXiv:1610.03403, Math. Z. 2018). Proves abundance for klt threefold pairs with Kodaira dimension one in characteristic $p>5$, an earlier positive-characteristic case close to the final step from $\kappa=1$ to semiampleness; it is not in this paper's bibliography.
Digests and expositions
- AI Proved What?, Abundance mod p? Threefolds in numerical dimension one (Episode 035, October 2026). This video and accompanying transcript outline the terminal paper's cycle, jet and Cartier-descent argument and the log-canonical paper's reduction to its companion's cycle obstruction. [AI-generated per source] [family-level]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 035
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Log-abundance-in-numerical-dimension-one-for-threefolds-in-positive-characteristic-October-5-2026,
author = {{OpenAI}},
title = {{Log abundance in numerical dimension one for threefolds in positive characteristic}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Log-abundance-in-numerical-dimension-one-for-threefolds-in-positive-characteristic-October-5-2026/paper.pdf}{OAI:Log-abundance-in-numerical-dimension-one-for-threefolds-in-positive-characteristic-October-5-2026}},
year = {2026}
}Family 035: Log-canonical threefold abundance in numerical dimension one
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. The adjoint is semiample, without requiring the original variety to be terminal or ℚ-factorial.
Other papers in this family: