Terminal 3-folds that are not Cohen-Macaulay

TL;DR

This paper demonstrates that terminal 3-fold singularities, which are typically Cohen-Macaulay in characteristic zero, can fail to be Cohen-Macaulay in characteristic p or mixed characteristic for p=2,3,5, with examples involving cyclic quotients.

Contribution

It provides explicit examples of non-Cohen-Macaulay terminal 3-folds in positive and mixed characteristic, highlighting the failure of a key vanishing theorem.

Findings

Examples of terminal 3-folds not Cohen-Macaulay in characteristic p or mixed characteristic.

Cyclic quotients of regular schemes can exhibit diverse singular behaviors in positive characteristic.

A new criterion for quotients to have toric singularities was developed.

Abstract

An important local vanishing theorem for the minimal model program is the fact that klt singularities in characteristic zero are Cohen-Macaulay. In contrast, even in the narrow setting of terminal singularities of dimension 3, we show that Cohen-Macaulayness can fail in characteristic $p$ or mixed characteristic $(0,p)$ for $p$ equal to 2, 3, or 5. This is optimal, by work of Arvidsson-Bernasconi-Lacini. The examples are quotients of regular schemes by the cyclic group $G$ of order $p$. In characteristic $p$ or mixed characteristic, such quotients can exhibit a wide range of behavior. Our key technical tool is a sufficient condition for quotients by $G$ to have only toric singularities.