Toricity in families of Fano varieties

TL;DR

This paper investigates the behavior of certain geometric properties, like being toric or of cluster type, within families of Fano varieties, showing these properties are constructible and finite in number.

Contribution

It proves that being toric or of cluster type is a constructible property in families of Fano varieties, and establishes finiteness results for smooth cluster type Fano varieties.

Findings

Being toric is a constructible property in families of Fano varieties.

Being of cluster type is a constructible property in families of $ ext{Q}$-factorial terminal Fano varieties.

There are finitely many smooth families parametrizing $n$-dimensional smooth cluster type Fano varieties.

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.