Three-dimensional isolated quotient singularities in even characteristic

TL;DR

This paper investigates three-dimensional quotient singularities in characteristic 2, proving conditions under which such quotients are non-singular and connecting the classification of these singularities to existing classifications in non-modular cases.

Contribution

It extends the understanding of modular quotient singularities in three dimensions, generalizing results to reducible groups and linking to non-modular classifications.

Findings

Quotients are non-singular under specified conditions.

Generalization of Kemper and Malle's theorem to reducible groups.

Reduction of classification problem to Vincent's non-modular cases.

Abstract

This paper is a complement to the work of the second author on modular quotient singularities in odd characteristic (see arXiv:1210.8006). Here we prove that if $V$ is a three-dimensional vector space over a field of characteristic $2$ and $G<GL(V)$ is a finite subgroup generated by pseudoreflections and possessing a $2$-dimensional invariant subspace $W$ such that the restriction of $G$ to $W$ is isomorphic to the group $SL_{2}(\mathbb{F}_{2^n})$, then the quotient $V/G$ is non-singular. This, together with earlier known results on modular quotient singularities, implies first that a theorem of Kemper and Malle on irreducible groups generated by pseudoreflections generalizes to reducible groups in dimension three, and, second, that the classification of three-dimensional isolated singularities which are quotients of a vector space by a linear finite group reduces to Vincent's classification of non-modular isolated quotient singularities.