Cluster algebras give rise to a class of Gorenstein rings which enjoy a large amount of symmetry. Concentrating on the rank 2 cases, we show how cluster varieties can be used to construct many interesting projective algebraic varieties. Our main application is then to construct hundreds of families of Fano 3-folds in codimensions 4 and 5. In particular, for Fano 3-folds in codimension 4 we construct at least one family for 187 of the 206 possible Hilbert polynomials contained in the Graded Ring Database.

中文翻译：

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从等级 2 的簇变体构造 Fano 3-fold

簇代数产生了一类具有大量对称性的 Gorenstein 环。专注于排名 2 的案例，我们展示了如何使用集群变体来构造许多有趣的射影代数变体。然后，我们的主要应用是在余维 4 和 5 中构造数百个 Fano 3-fold 族。特别是，对于余维 4 中的 Fano 3-fold，我们为 Graded 中包含的 206 个可能的 Hilbert 多项式中的 187 个构造至少一个族。环数据库。