Generalized frieze varieties and Gr\"obner bases

TL;DR

This paper investigates the structure of generalized frieze varieties associated with affine quivers, revealing their geometric properties and providing algorithms and Gr"obner bases for their defining equations.

Contribution

It introduces new results on the geometric nature of generalized frieze varieties for affine quivers and offers algorithms and Gr"obner bases for their explicit description.

Findings

Generalized frieze varieties are either finite sets or unions of rational curves.

If the dimension is one, each component has genus zero.

The paper provides algorithms and Gr"obner bases for defining polynomials.

Abstract

We study properties of generalized frieze varieties for quivers associated to cluster automorphisms. Special cases include acyclic quivers with Coxeter automorphisms and quivers with Cluster DT automorphisms. We prove that the generalized frieze variety X of an affine quiver with the Coxeter automorphism is either a finite set of points or a union of finitely many rational curves. In particular, if dim X=1, the genus of each irreducible component is zero. We also propose an algorithm to obtain the defining polynomials for each irreducible component of the generalized frieze variety for affine quivers. Furthermore, we give the Gr\"obner basis with respect to a given monomial order for each irreducible component of frieze varieties of affine quivers with given orientations, and show that each component is a smooth rational curve.