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arXiv · 2409.03564

Toricity in families of Fano varieties

Lena Ji·Joaquín Moraga
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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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Lena Ji, Joaquín Moraga. 2024-09-05. Toricity in families of Fano varieties. https://arxiv.org/abs/2409.03564

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