A Note on $\mathbb{Q}$-Gorenstein surfaces

TL;DR

This paper constructs a specific example of a $Q$-Gorenstein surface with a non-finitely generated canonical ring and provides a counterexample to the minimal model program in positive characteristic.

Contribution

It presents the first known counterexample to the minimal model program for $Q$-Gorenstein surfaces in positive characteristic.

Findings

Constructed a $Q$-Gorenstein surface with non-finitely generated canonical ring

Provided the first counterexample to the minimal model program in positive characteristic

Demonstrated limitations of existing classification theories in positive characteristic

Abstract

We construct a normal projective $\mathbb{Q}$-Gorenstein surface over an algebraically closed field whose canonical ring is not finitely generated. Moreover, we provide a counterexample to the minimal model program for $\mathbb{Q}$-Gorenstein surfaces, which was previously unknown in positive characteristic.