Classification of singularities of cluster algebras of finite type: the case of trivial coefficients

TL;DR

This paper classifies the singularities of finite type cluster algebras with trivial coefficients and develops a method for their desingularization, also exploring related non-finite type cluster algebras from star-shaped quivers.

Contribution

It provides a complete classification of singularities and a constructive desingularization method for finite type cluster algebras with trivial coefficients.

Findings

Complete classification of singularities for finite type cluster algebras

Constructive desingularization via blowups in regular centers

Analysis of non-finite type cluster algebras from star-shaped quivers

Abstract

We provide a complete classification of the singularities of cluster algebras of finite type with trivial coefficients. Alongside, we develop a constructive desingularization of these singularities via blowups in regular centers over fields of arbitrary characteristic. Furthermore, from the same perspective, we study a family of cluster algebras which are not of finite type and which arise from a star shaped quiver.