Arithmetic finiteness of Mukai varieties of genus 7
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915526303809536 |
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| author | Ito, Tetsushi Kanemitsu, Akihiro Takamatsu, Teppei Tanaka, Yuuji |
| author_facet | Ito, Tetsushi Kanemitsu, Akihiro Takamatsu, Teppei Tanaka, Yuuji |
| contents | We study arithmetic finiteness of prime Fano threefolds of genus 7 and their higher dimensional generalization, called Mukai varieties of genus 7. For prime Fano threefolds of genus 7, we provide an arithmetic refinement of the Torelli theorem, obtain Shafarevich-type finiteness results, and show the failure of the Néron--Ogg--Shafarevich criterion of good reduction. For Mukai varieties of genus 7, we prove that Shafarevich-type finiteness results hold in dimensions 9 and 10, but fail in dimension 6. In addition, we show that Mukai $n$-folds of genus 7 over $\mathbb{Z}$ do not exist for $n \leq 4$, whereas they exist for $5 \leq n \leq 10$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_20046 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Arithmetic finiteness of Mukai varieties of genus 7 Ito, Tetsushi Kanemitsu, Akihiro Takamatsu, Teppei Tanaka, Yuuji Algebraic Geometry Number Theory 11G35, 14J45, 14C34 We study arithmetic finiteness of prime Fano threefolds of genus 7 and their higher dimensional generalization, called Mukai varieties of genus 7. For prime Fano threefolds of genus 7, we provide an arithmetic refinement of the Torelli theorem, obtain Shafarevich-type finiteness results, and show the failure of the Néron--Ogg--Shafarevich criterion of good reduction. For Mukai varieties of genus 7, we prove that Shafarevich-type finiteness results hold in dimensions 9 and 10, but fail in dimension 6. In addition, we show that Mukai $n$-folds of genus 7 over $\mathbb{Z}$ do not exist for $n \leq 4$, whereas they exist for $5 \leq n \leq 10$. |
| title | Arithmetic finiteness of Mukai varieties of genus 7 |
| topic | Algebraic Geometry Number Theory 11G35, 14J45, 14C34 |
| url | https://arxiv.org/abs/2409.20046 |