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

Nao Moriyama

TL;DR

The paper demonstrates that the canonical ring of a normal projective $Q$-Gorenstein surface need not be finitely generated in positive characteristic and that the MMP may fail to preserve $Q$-Gorensteinness in this setting. It provides an algebraic construction of counterexamples, extending Sakai’s complex-analytic results to arbitrary characteristic, and proves a κ=2 finite-generation criterion linked to $Q$-Gorensteinness. It also shows that finite generation holds under favorable conditions (Gorenstein, $Q$-factorial, or κ≤1) and explains how to generate κ=2 counterexamples from MMP pathologies. Collectively, these results reveal limitations of finiteness and the MMP for $Q$-Gorenstein surfaces in positive characteristic and demonstrate the persistence of pathologies beyond characteristic zero.

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.

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

TL;DR

The paper demonstrates that the canonical ring of a normal projective -Gorenstein surface need not be finitely generated in positive characteristic and that the MMP may fail to preserve -Gorensteinness in this setting. It provides an algebraic construction of counterexamples, extending Sakai’s complex-analytic results to arbitrary characteristic, and proves a κ=2 finite-generation criterion linked to -Gorensteinness. It also shows that finite generation holds under favorable conditions (Gorenstein, -factorial, or κ≤1) and explains how to generate κ=2 counterexamples from MMP pathologies. Collectively, these results reveal limitations of finiteness and the MMP for -Gorenstein surfaces in positive characteristic and demonstrate the persistence of pathologies beyond characteristic zero.

Abstract

We construct a normal projective -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 -Gorenstein surfaces, which was previously unknown in positive characteristic.
Paper Structure (5 sections, 11 theorems, 41 equations)

This paper contains 5 sections, 11 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A criterion of finite generation
  4. Counterexamples to the MMP
  5. Counterexamples to the finite generation

Key Result

Theorem 1.2

Assume that $k \neq \overline{\mathbb{F}}_p$. Then, there exists a normal projective $\mathbb{Q}$-Gorenstein surface $X$ over $k$ such that $\kappa(X) = 2$ but the canonical ring $R(X)$ is not finitely generated.

Theorems & Definitions (24)

  • Theorem 1.2: cf. Section \ref{['sec: f.g.']}
  • Theorem 1.4: cf. Section \ref{['sec: MMP']}
  • Definition 2.1: Gorenstein, $\mathbb{Q}$-Gorenstein, or $\mathbb{Q}$-factorial surfaces
  • Remark 2.2: Numerical properties of Weil divisors in the sense of Mumford
  • Definition 2.3: Canonical rings and Kodaira dimension (cf. Sak87)
  • Definition 2.4
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • Theorem 3.3: cf. Sak87
  • ...and 14 more