# Non-Cohen-Macaulay canonical singularities

## TL;DR

This paper demonstrates the existence of non-Cohen-Macaulay canonical singularities in characteristic two by constructing a smooth Fano variety where the anti-canonical bundle's square violates Kodaira vanishing.

## Contribution

It introduces a novel example of canonical singularities that are non-Cohen-Macaulay in characteristic two, expanding understanding of singularities in algebraic geometry.

## Key findings

- Existence of smooth Fano varieties in characteristic two with Kodaira vanishing failure
- Construction of non-Cohen-Macaulay canonical singularities in characteristic two
- Violation of Kodaira vanishing for the square of the anti-canonical bundle

## Abstract

The purpose of this note is to show that in characteristic two there exists a smooth Fano variety for which the square of the anti-canonical bundle violates Kodaira vanishing. This is used in turn to construct non-Cohen-Macaulay canonical singularities in characteristic two.