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arXiv · 2409.20046

Arithmetic finiteness of Mukai varieties of genus 7

Tetsushi Ito·Akihiro Kanemitsu·Teppei Takamatsu·Yuuji Tanaka
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Abstract

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 N\'eron--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 $5 \leq n \leq 10$.

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Tetsushi Ito, Akihiro Kanemitsu, Teppei Takamatsu, Yuuji Tanaka. 2024-09-30. Arithmetic finiteness of Mukai varieties of genus 7. https://arxiv.org/abs/2409.20046

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