math.AC — Commutative Algebra

Counterexamples in positive characteristics

Contributed by Rahul Ajit, Seungsu Lee

AI found the following counterexamples (following human directions) to 3 important questions: For every prime $p$ 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 $F$-finite splinter with regular punctured spectrum and embedding dimension six. Both test ideals are the maximal ideal, so the cone is not weakly $F$-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.