The minimal volume of stable surfaces of rank one

TL;DR

This paper determines the minimal volume of stable surfaces of rank one, proving uniqueness and introducing an AI-assisted plurigenus inequality as a key tool, with implications for classification problems in algebraic geometry.

Contribution

It establishes the minimal volume and uniqueness of stable surfaces of rank one, utilizing an AI-derived plurigenus inequality for the first time in this context.

Findings

Identified the minimal volume of stable surfaces of rank one.

Proved the uniqueness of the surface attaining this minimum.

Applied the AI-derived inequality to classify small-volume threefolds and solve Kollár's problem.

Abstract

We determine the minimal volume of a stable surface of rank one, and show that the surface attaining this minimum is unique up to isomorphism. This resolves a conjecture of Alexeev and the second author. Of independent interest, the decisive step of the proof uses a plurigenus inequality re-derived by an AI chatbot and applied as a pluricanonical filter; we further apply this filter to rule out additional cases in the classification of small-volume threefolds of general type, and in Koll\'ar's algebraic Montgomery--Yang problem. The underlying inequality has classical antecedents. To our knowledge this is the first paper in birational geometry to claim a C2-level human--AI collaboration in the sense of Feng et al., where the AI's contribution is the recognition that this inequality functions as the decisive pluricanonical filter in the basket analysis.