Plectis Atlas
Search

↑ and ↓ move, Enter opens, Esc closes.

Show on the map

Log-canonical threefold abundance in numerical dimension one

OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this 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.

The release has no Lean statement for this family, so there is no formal statement to compare with the result described above.

What the papers state

Theorem 1.1

In Log abundance in numerical dimension one for threefolds in positive characteristic, 5 October 2026

Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.

Let k be an algebraically closed field of characteristic p>3. Let (X,B) be a projective log canonical threefold pair over k, where X is normal, B is an effective \Q-divisor, and K_X+B is \Q-Cartier. If K_X+B is nef and has numerical dimension one, then K_X+B is semiample.

Theorem 1.1

In Abundance in numerical dimension one for terminal threefolds in positive characteristic, 24 September 2026

The paper’s main theorem is shown as a possible match to OpenAI’s description. No review confirming the selection is recorded.

Let X be a normal projective \Q-factorial threefold over an algebraically closed field k of characteristic p>3. Suppose that X has terminal singularities and that K_X is nef with \nu(K_X)=1. Then some positive Cartier multiple of K_X has at least two independent global sections. In particular, \kappa(X,K_X)=1 and K_X is semiample.

The release does not say which paper theorem its catalogue describes. Both statements above are shown whole, the closer match to the description’s wording first. The atlas has not reviewed either choice.

Reported proof verification
The release has no Lean statement for this family. The papers are available to read and assess.
Papers and explanations
Two current papers, each readable here as a page beside its PDF.

Papers

Each paper opens as a reading page in the browser, with the PDF beside it. The statements each paper opens with are on its reading page.

Log abundance in numerical dimension one for threefolds in positive characteristic

5 October 2026, current edition.

Read it herePDF

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.

Lean statements

The release has no Lean statement for this family. Its papers remain available to read, and contributors can propose a formal statement of a result. Suggest a change

Independent comparisons and proof-verification reports

Does the Lean express the advertised result?

The release contains no Lean statement for this family, so there is no formal statement to compare with OpenAI’s description.

What these papers cite in the release, and what cites them

These links come from the papers’ bibliographies. A citation records that one paper cites another; how each cited result is used has not been examined here.

Within this family

Papers elsewhere in the release that this family’s papers cite

This family’s papers cite no paper elsewhere in the release.

Papers elsewhere in the release that cite this family’s papers

No paper elsewhere in the release cites this family’s papers.

References outside the release

Each paper’s outside references are listed with it on its reading page, where each citation is placed. Every outside work cited in the release: References outside the release

Explain a result, compare statements or propose a correction

Atlas has no contributed explanation for this family yet. You can offer one with a GitHub account. You receive named credit and disclose any AI assistance.

Each button opens a GitHub form with this page and its source version filled in. Opening the form does not reserve the work.

Offer to write an explanationSuggest a source attributionSuggest a change

What people have said

Commentary on this family by others, dated and credited. It is commentary, not a finding.

  • An unsigned public tier list, the OpenAI Math Tier List, classes it as a proof and rates its significance, if it holds up, as “A strong professor’s paper of the year”. The page says that nothing on it is verified.

    Author not stated. Source

  • Fabio Bernasconi, Jakub Witaszek, Zheng Xu, Abundance for threefolds in positive characteristic, 7 October 2026. Source

    Bernasconi, Witaszek and Xu released a preliminary abundance theorem for threefolds in characteristic above 42, saying they did so after OpenAI’s list claimed the case of characteristic above 3.

Questions awaiting review

Each item identifies a statement, definition or report that needs further review. Its link opens a GitHub form with the source version already recorded.

Identify the theorem the catalogue describes

  • Family 035 The theorem shown was selected from shared wording with OpenAI’s description. Atlas records no review confirming that it is the theorem the description means. Open the form

State the result in Lean

  • Family 035 The release has no Lean statement for this family. Open the form