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| मुख्य लेखकों: | , , |
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| स्वरूप: | Preprint |
| प्रकाशित: |
2024
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| विषय: | |
| ऑनलाइन पहुंच: | https://arxiv.org/abs/2412.17033 |
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| _version_ | 1866914325101281280 |
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| author | Catanese, Fabrizio Liu, Wenfei Schütt, Matthias |
| author_facet | Catanese, Fabrizio Liu, Wenfei Schütt, Matthias |
| contents | In this second part we study first the group $Aut_{\mathbb Q}(S)$ of numerically trivial automorphisms of an algebraic properly elliptic surface $S$, that is, of a minimal algebraic surface with Kodaira dimension $κ(S)=1$, in the case $χ(S) \geq 1$. Our first surprising result is that, against what has been believed for over 40 years, there exist nontrivial such groups for $p_g(S) >0$. Indeed, we show even that $Aut_{\mathbb Q}(S)$ is always a 2-generated finite abelian group, but there is no absolute upper bound for its cardinality. At any rate, we give explicit and essentially optimal upper bounds for $|Aut_{\mathbb Q}(S)|$ in terms of the numerical invariants of $S$, as $χ(S)$, or the irregularity $q(S)$, or the bigenus $P_2(S)$. Moreover, we reach an almost complete description of the possible groups $Aut_{\mathbb Q}(S)$ and we give effective criteria for such surfaces to have trivial $Aut_{\mathbb Q}(S)$. Our second surprising results concern the quite elusive group $Aut_{\mathbb Z}(S)$ of cohomologically trivial automorphisms; we are able to give the explicit upper bounds for $|Aut_{\mathbb Z}(S)|$ in special cases: 9 when $p_g(S) =0$, and we achieve the sharp upper bound 3 when $S$ (i.e., the pluricanonical elliptic fibration) is isotrivial. Also in the non isotrivial case we produce subtle examples where $Aut_{\mathbb Z}(S)$ is a group of order 2 or 3. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_17033 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the numerically and cohomologically trivial automorphisms of elliptic surfaces II: $χ(S)>0$ Catanese, Fabrizio Liu, Wenfei Schütt, Matthias Algebraic Geometry Complex Variables 14J50, 14J80, 14J27, 14H30, 14F99, 32L05, 32M99, 32Q15, 32Q55 In this second part we study first the group $Aut_{\mathbb Q}(S)$ of numerically trivial automorphisms of an algebraic properly elliptic surface $S$, that is, of a minimal algebraic surface with Kodaira dimension $κ(S)=1$, in the case $χ(S) \geq 1$. Our first surprising result is that, against what has been believed for over 40 years, there exist nontrivial such groups for $p_g(S) >0$. Indeed, we show even that $Aut_{\mathbb Q}(S)$ is always a 2-generated finite abelian group, but there is no absolute upper bound for its cardinality. At any rate, we give explicit and essentially optimal upper bounds for $|Aut_{\mathbb Q}(S)|$ in terms of the numerical invariants of $S$, as $χ(S)$, or the irregularity $q(S)$, or the bigenus $P_2(S)$. Moreover, we reach an almost complete description of the possible groups $Aut_{\mathbb Q}(S)$ and we give effective criteria for such surfaces to have trivial $Aut_{\mathbb Q}(S)$. Our second surprising results concern the quite elusive group $Aut_{\mathbb Z}(S)$ of cohomologically trivial automorphisms; we are able to give the explicit upper bounds for $|Aut_{\mathbb Z}(S)|$ in special cases: 9 when $p_g(S) =0$, and we achieve the sharp upper bound 3 when $S$ (i.e., the pluricanonical elliptic fibration) is isotrivial. Also in the non isotrivial case we produce subtle examples where $Aut_{\mathbb Z}(S)$ is a group of order 2 or 3. |
| title | On the numerically and cohomologically trivial automorphisms of elliptic surfaces II: $χ(S)>0$ |
| topic | Algebraic Geometry Complex Variables 14J50, 14J80, 14J27, 14H30, 14F99, 32L05, 32M99, 32Q15, 32Q55 |
| url | https://arxiv.org/abs/2412.17033 |