Toricity in families of Fano varieties
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866908558684061696 |
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| author | Ji, Lena Moraga, Joaquín |
| author_facet | Ji, Lena Moraga, Joaquín |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_03564 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Toricity in families of Fano varieties Ji, Lena Moraga, Joaquín Algebraic Geometry 14M25 (Primary), 14E30, 14J10, 14J45 (Secondary) 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. |
| title | Toricity in families of Fano varieties |
| topic | Algebraic Geometry 14M25 (Primary), 14E30, 14J10, 14J45 (Secondary) |
| url | https://arxiv.org/abs/2409.03564 |