Generalized frieze varieties and Gr\"obner bases

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.