Arithmetic finiteness of Mukai varieties of genus 7

TL;DR

This paper investigates the arithmetic properties and finiteness results of Mukai varieties of genus 7, extending classical theorems and examining their existence across various dimensions.

Contribution

It provides new arithmetic finiteness results for Mukai varieties of genus 7, refining the Torelli theorem and analyzing good reduction failures across dimensions.

Findings

Finiteness results hold in dimensions 9 and 10 but not in 6.

Prime Fano threefolds of genus 7 satisfy Shafarevich-type finiteness.

Mukai n-folds of genus 7 do not exist over $\

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 $5 \leq n \leq 10$.