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<p>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 <i>X</i> 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 <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3820_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="81" /> </InlineMediaObject> <EquationSource Format="TEX">\(\dim X=1,\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>dim</mo> <mi>X</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> the genus of each irreducible component of <i>X</i> is zero. We also propose an algorithm to obtain the defining polynomials for each irreducible component of the generalized frieze variety <i>X</i> of an affine quiver. Furthermore, we give the Gröbner 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.</p>

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Generalized frieze varieties and Gröbner bases

  • 期刊论文

  • 发表日期：

    2025年7月23日

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  • Siyang Liu

摘要

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 \(\dim X=1,\) dim X = 1 , the genus of each irreducible component of X is zero. We also propose an algorithm to obtain the defining polynomials for each irreducible component of the generalized frieze variety X of an affine quiver. Furthermore, we give the Gröbner 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.