HEXAGONbeta

Counterexamples in positive characteristics

Primarily AI-generated textHuman understanding: some partsmath.AC — Commutative Algebramath.AG — Algebraic Geometry

Contributed by Rahul Ajit ↗, Seungsu Lee ↗

Version 1 / Nov 08, 2026 / CC BY 4.0

Abstract

AI found the following counterexamples (following human directions) to 3 important questions: For every prime pp we construct a smooth projective rational surface that is globally ++-regular, with anticanonical divisor nef but neither big nor semiample. Its completed cone is a three-dimensional FF-finite splinter with regular punctured spectrum and embedding dimension six. Both test ideals are the maximal ideal, so the cone is not weakly FF-regular. A discrepancy calculation by humans rules out every effective klt boundary. A separate four-dimensional complete normal local domain has distinct big and finitistic test ideals.

Author comments: All main ideas and constructions in this draft are either explicitly suggested or fully developed by AI. The humans (R. Ajit and S. Lee) asked specific questions and gave related directions, papers, and even precise result statements to AI so that it could explore those directions and construct counter-examples. In most cases, AIs (and humans) were successful. However, their mathematical details were quite incomprehensible to the humans. Currently, Ajit and Lee are carefully verifying AI's claims as well as improving the exposition and adding their personal human inputs. What follows is a very preliminary attempt to that. In particular, the humans think the math in this draft is correct. In the near future, with a better understanding, the humans think that they will be able to substantially improve the exposition, add and correct proper citations, and most importantly add precise human contributions (right definitions, for example) to this project. We will also add more precise human contributions to the existing material soon. We make this available to the world purely for mathematical reasons.

Provenance statement

The contributors asked specific questions, supplied specific references (in some cases, even pointed out specific results to use) in ChatGPT and Claude to get these examples. Later, we used ChatGPT and Claude to understand and simplify (like reducing complexity) and generalize (to all char p) certain constructions, as well as to prepare this draft.

Tools used

Anthropic
Claude Fable
OpenAI
ChatGPT

References

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Version history

  1. v1 · 42 pagesSubmitted by Rahul AjitAll main ideas and constructions in this draft are either explicitly suggested or fully developed by AI. The humans (R. Ajit and S. Lee) asked specific questions and gave related directions, papers, and even precise result statements to AI so that it could explore those directions and construct counter-examples. In most cases, AIs (and humans) were successful. However, their mathematical details were quite incomprehensible to the humans. Currently, Ajit and Lee are carefully verifying AI's claims as well as improving the exposition and adding their personal human inputs. What follows is a very preliminary attempt to that. In particular, the humans think the math in this draft is correct. In the near future, with a better understanding, the humans think that they will be able to substantially improve the exposition, add and correct proper citations, and most importantly add precise human contributions (right definitions, for example) to this project. We will also add more precise human contributions to the existing material soon. We make this available to the world purely for mathematical reasons.CurrentNov 08, 2026