A new equivalence between singularity categories of commutative algebras

TL;DR

This paper establishes a novel triangle equivalence between the singularity categories of two isolated cyclic quotient singularities of different Krull dimensions, revealing new connections in algebraic geometry.

Contribution

It introduces the first example of a singular equivalence linking connected commutative algebras of odd and even Krull dimensions.

Findings

Established a triangle equivalence between singularity categories of specific singularities

Connected singularity categories of different Krull dimensions for the first time

Extended the equivalence to certain quasi-projective varieties via Orlov's result

Abstract

We construct a triangle equivalence between the singularity categories of two isolated cyclic quotient singularities of Krull dimensions two and three, respectively. This is the first example of a singular equivalence involving connected commutative algebras of odd and even Krull dimension. In combination with Orlov's localization result, this gives further singular equivalences between certain quasi-projective varieties of dimensions two and three, respectively.