Algebraic and complex geometry · Result 035

Log-canonical threefold abundance in numerical dimension one

2 papers

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.

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▶Introduction: the problem and why it mattersin preparation

▶The presented solutionin preparation

Papers 2

Log abundance in numerical dimension one for threefolds in positive characteristic

OpenAI · October 5, 2026

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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@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}
}

Abundance in numerical dimension one for terminal threefolds in positive characteristic

OpenAI · September 24, 2026

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 K_X is nef with \(\nu(K_X)=1\), then K_X is semiample and \(\kappa(X,K_X)=1\).

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@misc{OAI:Abundance-in-numerical-dimension-one-for-terminal-threefolds-in-positive-characteristic-September-24-2026,
  author = {{OpenAI}},
  title = {{Abundance in numerical dimension one for terminal threefolds in positive characteristic}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Abundance-in-numerical-dimension-one-for-terminal-threefolds-in-positive-characteristic-September-24-2026/paper.pdf}{OAI:Abundance-in-numerical-dimension-one-for-terminal-threefolds-in-positive-characteristic-September-24-2026}},
  year = {2026}
}

Formal verification

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