On regular surfaces of general type with numerically trivial automorphism group of order $4$
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929644082561024 |
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| author | Cai, Jin-Xing Liu, Wenfei |
| author_facet | Cai, Jin-Xing Liu, Wenfei |
| contents | Let $S$ be a regular minimal surface of general type over the field of complex numbers, and $\mathrm{Aut}_\mathbb{Q}(S)$ the subgroup of automorphisms acting trivially on $H^*(S,\mathbb{Q})$. It has been known since twenty years that $|\mathrm{Aut}_\mathbb{Q}(S)|\leq 4$ if the invariants of $S$ are sufficiently large. Under the assumption that $K_S$ is ample, we characterize the surfaces achieving the equality, showing that they are isogenous to a product of two curves, of unmixed type, and that the group $\mathrm{Aut}_\mathbb{Q}(S)$ is isomorphic to $(\mathbb{Z}/2\mathbb{Z})^2$. Moreover, unbounded families of surfaces with $\mathrm{Aut}_\mathbb{Q}(S)\cong(\mathbb{Z}/2\mathbb{Z})^2$ are provided. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_16501 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On regular surfaces of general type with numerically trivial automorphism group of order $4$ Cai, Jin-Xing Liu, Wenfei Algebraic Geometry 14J50, 14J29 Let $S$ be a regular minimal surface of general type over the field of complex numbers, and $\mathrm{Aut}_\mathbb{Q}(S)$ the subgroup of automorphisms acting trivially on $H^*(S,\mathbb{Q})$. It has been known since twenty years that $|\mathrm{Aut}_\mathbb{Q}(S)|\leq 4$ if the invariants of $S$ are sufficiently large. Under the assumption that $K_S$ is ample, we characterize the surfaces achieving the equality, showing that they are isogenous to a product of two curves, of unmixed type, and that the group $\mathrm{Aut}_\mathbb{Q}(S)$ is isomorphic to $(\mathbb{Z}/2\mathbb{Z})^2$. Moreover, unbounded families of surfaces with $\mathrm{Aut}_\mathbb{Q}(S)\cong(\mathbb{Z}/2\mathbb{Z})^2$ are provided. |
| title | On regular surfaces of general type with numerically trivial automorphism group of order $4$ |
| topic | Algebraic Geometry 14J50, 14J29 |
| url | https://arxiv.org/abs/2412.16501 |