Arithmetic finiteness of Mukai varieties of genus 7

TL;DR

The paper develops an arithmetic theory for Mukai varieties of genus $7$, proving an arithmetic Torelli theorem and a cohomological Shafarevich finiteness result for prime Fano threefolds of genus $7$, while also producing a counterexample to the Néron--Ogg--Shafarevich criterion. It extends finiteness results to Mukai $n$-folds in higher dimensions, showing Shafarevich finiteness for $n=9,10$ but failure for $n=6$, and determining over ${\mathbb Z}$ which Mukai $n$-folds exist ($n\le 4$ do not, $5\le n\le 10$ do). The work leverages spinor tenfold geometry, duality between genus $7$ curves and prime Fano threefolds, and mixed-characteristic two-ray techniques to connect geometric and arithmetic data via intermediate Jacobians and their algebraic representatives. These results illuminate the arithmetic behavior of Picard rank $1$ Fano/Mukai varieties, reveal new instances where Shafarevich finiteness interacts with integral models, and relate to broader questions about K-stability and moduli in arithmetic geometry.

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 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 $5 \leq n \leq 10$.

Theorems & Definitions (106)

• Theorem 1.1: Arithmetic Torelli theorem; see Theorem \ref{['thm:moduliarithmetictorelli']}
• Remark 1.2
• Theorem 1.3: Cohomological Shafarevich conjecture; see Theorem \ref{['thm:cohomshaf']}
• Corollary 1.4: Ordinary Shafarevich conjecture
• Remark 1.5
• Theorem 1.6: Counter-examples to the Néron--Ogg--Shafarevich criterion; see Theorem \ref{['thm:grcounterexample']}
• Remark 1.7
• Theorem 1.8: Ordinary Shafarevich conjecture for Mukai $n$-folds of genus $7$
• Remark 1.9
• Theorem 1.10: Fontaine-type results for Mukai $n$-folds of genus $7$; see Theorem \ref{['thm:MukainoverZ']}