Automorphisms of Generalized Fermat manifolds
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2020
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| _version_ | 1866916523709300736 |
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| author | Hidalgo, Ruben A. Hughes, Henry F. Leyton-Alvarez, Maximiliano |
| author_facet | Hidalgo, Ruben A. Hughes, Henry F. Leyton-Alvarez, Maximiliano |
| contents | Let $d \geq 1$, $k \geq 2$ and $n\geq d+1$ be integers. A $d$-dimensional smooth complex algebraic variety $M$ is called a generalized Fermat variety of type $(d;k,n)$ if there is a Galois holomorphic branched covering $π:M \to {\mathbb P}^{d}$, with deck group $H\cong {\mathbb Z}_{k}^{n}$, whose branch divisor consists of $n+1$ hyperplanes in general position, each one of branch order $k$. In this case, $H$ is called a generalized Fermat group of type $(d;k,n)$. In previous work, we proved that the generalized Fermat group $H$ is unique in the following cases: (i) $d=1$ and $(k-1)(n-1)>2$, or (ii) $d \geq 2$ and $(d;k,n) \notin \{(2;2,5), (2;4,3)\}$. To obtain this uniqueness fact, we used a differential method due to Kontogeorgis. This paper provides a different and shorter proof of the uniqueness of $H$. We also study the locus of fixed points of subgroups of $H$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2010_04628 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Automorphisms of Generalized Fermat manifolds Hidalgo, Ruben A. Hughes, Henry F. Leyton-Alvarez, Maximiliano Algebraic Geometry Complex Variables 14J50, 32Q40, 53C15 Let $d \geq 1$, $k \geq 2$ and $n\geq d+1$ be integers. A $d$-dimensional smooth complex algebraic variety $M$ is called a generalized Fermat variety of type $(d;k,n)$ if there is a Galois holomorphic branched covering $π:M \to {\mathbb P}^{d}$, with deck group $H\cong {\mathbb Z}_{k}^{n}$, whose branch divisor consists of $n+1$ hyperplanes in general position, each one of branch order $k$. In this case, $H$ is called a generalized Fermat group of type $(d;k,n)$. In previous work, we proved that the generalized Fermat group $H$ is unique in the following cases: (i) $d=1$ and $(k-1)(n-1)>2$, or (ii) $d \geq 2$ and $(d;k,n) \notin \{(2;2,5), (2;4,3)\}$. To obtain this uniqueness fact, we used a differential method due to Kontogeorgis. This paper provides a different and shorter proof of the uniqueness of $H$. We also study the locus of fixed points of subgroups of $H$. |
| title | Automorphisms of Generalized Fermat manifolds |
| topic | Algebraic Geometry Complex Variables 14J50, 32Q40, 53C15 |
| url | https://arxiv.org/abs/2010.04628 |