Result 035, Algebraic and complex geometry

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.

Proof

The bigger picture

Why it matters

An adjoint divisor combines a variety's canonical divisor, which encodes top-degree differential forms, with weighted boundary hypersurfaces. The manuscript reports when weak numerical positivity forces this object to define an actual geometric map for three-dimensional varieties.

What changes?

For projective threefolds over algebraically closed fields of characteristic p greater than 3, the manuscript assumes a log canonical variety-boundary pair, allowing controlled singularities, with nonnegative rational boundary weights. The adjoint must be rationally Cartier, so some positive multiple defines a line bundle; nef, so it intersects every curve nonnegatively; and of numerical dimension one, expressing one-dimensional numerical positivity. It reports semiampleness: some positive multiple has global sections with no common zero. Neither terminality nor rational factoriality is required.

What does that help mathematicians do?

Semiampleness turns numerical information about intersections into a map defined everywhere on the threefold. Here numerical dimension one means that the map's image is a curve. Researchers can therefore study these threefolds through their fibers over that curve, rather than only through intersection numbers. The claim also removes the terminal and rational factoriality assumptions imposed in the supplied earlier result, while retaining the dimension and characteristic restrictions.

Are there practical applications?

The immediate value is foundational: the result connects numerical positivity with geometric structure in the classification of algebraic threefolds in positive characteristic. It supplies an abundance statement for pairs with boundaries and log canonical singularities. Its reach remains specific to numerical dimension one; it does not establish abundance in other numerical dimensions or characteristics.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

2 manuscripts

Log abundance in numerical dimension one for threefolds in positive characteristic

October 5, 2026 16 pages

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.

PDF LaTeX source

Cite (BibTeX)

@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

September 24, 2026 46 pages

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 $\kappa(X,K_X)=1$ .

PDF LaTeX source

Cite (BibTeX)

@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}
}

At a glance

Subject: Algebraic and complex geometry
Manuscripts: 2
Pages: 62
Dated: Sep 24 to Oct 5, 2026
Lean formalization: No
Reasoning summary: No

Named problems addressed

Indexed by Emergent Mind with an LLM from the text above; check the manuscripts for the precise statements.

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