Toricity in families of Fano varieties

2024/09/05 by Lena Ji, Ji, Lena, Joaquí­n Moraga +1 · 3 citations
Social Sciences · #14E30 #14J10 #14J45 (Secondary) #14M25 (Primary) #African history and culture studies #Algebraic Geometry (math.AG) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2409.03564

openalex publication_date 2024/09/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Rationality is not a constructible property in families. In this article, we consider stronger notions of rationality and study their behavior in families of Fano varieties. We first show that being toric is a constructible property in families of Fano varieties. The second main result of this article concerns an intermediate notion that lies between toric and rational varieties, namely cluster type varieties. A cluster type \mathbb Q-factorial Fano variety contains an open dense algebraic torus, but the variety does not need to be endowed with a torus action. We prove that, in families of \mathbb Q-factorial terminal Fano varieties, being of cluster type is a constructible condition. As a consequence, we show that there are finitely many smooth families parametrizing n-dimensional smooth cluster type Fano varieties.

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• Fano compactifications of mutation algebras
• Degenerations of cluster type varieties
• Geometry of tropical mutation surfaces with a single mutation

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