On a Torelli Principle for automorphisms of Klein hypersurfaces
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866914758570016768 |
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| author | González-Aguilera, Víctor Liendo, Alvaro Montero, Pedro Loyola, Roberto Villaflor |
| author_facet | González-Aguilera, Víctor Liendo, Alvaro Montero, Pedro Loyola, Roberto Villaflor |
| contents | Using a refinement of the differential method introduced by Oguiso and Yu, we provide effective conditions under which the automorphisms of a smooth degree $d$ hypersurface of $\mathbf{P}^{n+1}$ are given by generalized triangular matrices. Applying this criterion we compute all the remaining automorphism groups of Klein hypersurfaces of dimension $n\geq 1$ and degree $d\geq 3$ with $(n,d)\neq (2,4)$. We introduce the concept of extremal polarized Hodge structures, which are structures that admit an automorphism of large prime order. Using this notion, we compute the automorphism group of the polarized Hodge structure of certain Klein hypersurfaces that we call of Wagstaff type, which are characterized by the existence of an automorphism of large prime order. For cubic hypersurfaces and some other values of $(n,d)$, we show that both groups coincide (up to involution) as predicted by the Torelli Principle. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2212_13308 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On a Torelli Principle for automorphisms of Klein hypersurfaces González-Aguilera, Víctor Liendo, Alvaro Montero, Pedro Loyola, Roberto Villaflor Algebraic Geometry 14C30, 14C34, 14J50, 14J70 Using a refinement of the differential method introduced by Oguiso and Yu, we provide effective conditions under which the automorphisms of a smooth degree $d$ hypersurface of $\mathbf{P}^{n+1}$ are given by generalized triangular matrices. Applying this criterion we compute all the remaining automorphism groups of Klein hypersurfaces of dimension $n\geq 1$ and degree $d\geq 3$ with $(n,d)\neq (2,4)$. We introduce the concept of extremal polarized Hodge structures, which are structures that admit an automorphism of large prime order. Using this notion, we compute the automorphism group of the polarized Hodge structure of certain Klein hypersurfaces that we call of Wagstaff type, which are characterized by the existence of an automorphism of large prime order. For cubic hypersurfaces and some other values of $(n,d)$, we show that both groups coincide (up to involution) as predicted by the Torelli Principle. |
| title | On a Torelli Principle for automorphisms of Klein hypersurfaces |
| topic | Algebraic Geometry 14C30, 14C34, 14J50, 14J70 |
| url | https://arxiv.org/abs/2212.13308 |