Fano threefolds of genus 12 with large automorphism group in positive and mixed characteristic

TL;DR

This paper classifies prime Fano threefolds of genus 12 with large automorphism groups over various fields, exploring their existence, automorphism structures, and arithmetic properties in positive and mixed characteristic settings.

Contribution

It provides a classification of $V_{22}$-varieties with large automorphism groups over perfect fields, including existence criteria and arithmetic implications.

Findings

Existence of $V_{22}$-varieties over fields with char ≠ 2,5

Shafarevich conjecture holds for certain types

Non-existence over $ ext{Spec}(bZ)$ with positive-dimensional automorphisms

Abstract

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.