Arithmetic finiteness of Mukai varieties of genus 7

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 $5 \leq n \leq 10$.