Weakly-exceptional singularities in higher dimensions

TL;DR

This paper demonstrates the existence of infinitely many Gorenstein weakly-exceptional quotient singularities in all dimensions, provides criteria for five- and six-dimensional cases, and connects these findings to classical geometric constructions.

Contribution

It introduces new existence results and criteria for weakly-exceptional quotient singularities in higher dimensions, linking algebraic geometry with singularity theory.

Findings

Existence of infinitely many Gorenstein weakly-exceptional quotient singularities in all dimensions.

A weak-exceptionality criterion for five-dimensional quotient singularities.

A sufficient condition for weakly-exceptionality in six-dimensional quotient singularities.

Abstract

We show that infinitely many Gorenstein weakly-exceptional quotient singularities exist in all dimensions, we prove a weak-exceptionality criterion for five-dimensional quotient singularities, and we find a sufficient condition for being weakly-exceptional for six-dimensional quotient singularities. The proof is naturally linked to various classical geometrical constructions related to subvarieties of small degree in projective spaces, in particular Bordiga surfaces and Bordiga threefolds.