Boundedness of total Cartier indices for rational singularities in families

TL;DR

This paper proves that in bounded families of varieties with rational singularities, the total Cartier index remains bounded, addressing a problem posed by Han and Jiang.

Contribution

It establishes a uniform bound on the Cartier indices for rational singularities in bounded families, with a novel proof structure partly generated by AI.

Findings

Boundedness of the total Cartier index in families of rational singularities.

The proof treats surface and higher-dimensional cases separately.

The proof structure was initially generated by AI and then refined by hand.

Abstract

We show that the total Cartier index of varieties with rational singularities in a bounded family is bounded. This solves a problem of Han and Jiang. The overall structure of the proof, which treats the surface case and the higher-dimensional case separately, was originated by generative AI, particularly the Rethlas system, and was substantially corrected and elaborated by hand.