Fano threefolds of genus 12 with large automorphism group in positive and mixed characteristic
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
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2026
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| _version_ | 1866911377173512192 |
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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 prime Fano threefolds of genus 12 ($V_{22}$-varieties) with positive-dimensional automorphism groups in positive and mixed characteristic. We classify such varieties over any perfect field. In particular, we prove that $V_{22}$-varieties of Mukai-Umemura type over $k$ exist if and only if $\mathrm{char}\ k \neq 2$, $5$. We also prove the same result for $\mathbb{G}_a$-type. As arithmetic applications, we show that the Shafarevich conjecture holds for $V_{22}$-varieties of Mukai-Umemura type and of $\mathbb{G}_m$-type, while it fails for $V_{22}$-varieties of $\mathbb{G}_a$-type. Moreover, we prove that there exists $V_{22}$-varieties over $\mathbb{Z}$, whereas there do not exist $V_{22}$-varieties over $\mathbb{Z}$ whose generic fiber has a positive-dimensional automorphism group. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_10106 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Fano threefolds of genus 12 with large automorphism group in positive and mixed characteristic Ito, Tetsushi Kanemitsu, Akihiro Takamatsu, Teppei Tanaka, Yuuji Algebraic Geometry Number Theory 11G35, 14J45, 14G17 We study prime Fano threefolds of genus 12 ($V_{22}$-varieties) with positive-dimensional automorphism groups in positive and mixed characteristic. We classify such varieties over any perfect field. In particular, we prove that $V_{22}$-varieties of Mukai-Umemura type over $k$ exist if and only if $\mathrm{char}\ k \neq 2$, $5$. We also prove the same result for $\mathbb{G}_a$-type. As arithmetic applications, we show that the Shafarevich conjecture holds for $V_{22}$-varieties of Mukai-Umemura type and of $\mathbb{G}_m$-type, while it fails for $V_{22}$-varieties of $\mathbb{G}_a$-type. Moreover, we prove that there exists $V_{22}$-varieties over $\mathbb{Z}$, whereas there do not exist $V_{22}$-varieties over $\mathbb{Z}$ whose generic fiber has a positive-dimensional automorphism group. |
| title | Fano threefolds of genus 12 with large automorphism group in positive and mixed characteristic |
| topic | Algebraic Geometry Number Theory 11G35, 14J45, 14G17 |
| url | https://arxiv.org/abs/2601.10106 |