Canonical singularities of dimension three in characteristic 2 which do not follow Reid's rules

TL;DR

This paper explores unique three-dimensional canonical singularities in characteristic 2 that defy Reid's rules, providing concrete examples and aiming to extend Reid's classification to positive characteristic fields.

Contribution

It presents new examples of compound Du Val singularities in characteristic 2 with one-dimensional singular loci that cannot be decomposed into products, advancing the classification of canonical singularities in positive characteristic.

Findings

Existence of numerous examples with rational double points of type D

Singularities with one-dimensional loci that are not product decompositions

Extension of Reid's classification to characteristic 2

Abstract

We continue to study and present concrete examples in characteristic 2 of compound Du Val singularities defined over an algebraically closed field which have one dimensional singular loci but cannot be written as products (a rational double point) x (a curve) up to analytic isomorphism at any point of the loci. Unlike in other characteristics, we find a large number of such examples whose general hyperplane sections have rational double points of type D. We consider these compound Du Val singularities as a special class of canonical singularities, intend to complete classification in arbitrary characteristic reinforcing Miles Reid's result in characteristic zero.