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

Tetsushi Ito, Akihiro Kanemitsu, Teppei Takamatsu, Yuuji Tanaka

TL;DR

The authors classify prime Fano threefolds of genus $12$ with nontrivial automorphism groups over perfect fields, distinguishing Mukai–Umemura, $\mathbb{G}_a$-type, and $\mathbb{G}_m$-type $V_{22}$-varieties and establishing characteristic-dependent existence and nonexistence results. They develop a relative two-ray game linking $V_{22}$-varieties to the quintic del Pezzo $V_5$-varieties and analyze quintic curves with large stabilizers to achieve a complete classification in positive and mixed characteristics; they also study degenerations and liftability to Witt rings. Arithmetic consequences include finite Shafarevich sets for Mukai–Umemura and $\mathbb{G}_m$-types, an infinite Shafarevich set for $\mathbb{G}_a$-type, and a detailed description of good reduction phenomena, including unit-equation and Br- and cohomology-based criteria. They prove the existence of a $V_{22}$-variety over $\mathbb{Z}$ but show that no such variety over $\mathbb{Z}$ has a generically positive-dimensional automorphism group, tying geometric classification to arithmetic constraints. These results significantly extend the understanding of genus-$12$ Fano threefolds in positive/mixed characteristic and illuminate the interplay between geometry, deformation theory, and arithmetic.

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.

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

TL;DR

The authors classify prime Fano threefolds of genus with nontrivial automorphism groups over perfect fields, distinguishing Mukai–Umemura, -type, and -type -varieties and establishing characteristic-dependent existence and nonexistence results. They develop a relative two-ray game linking -varieties to the quintic del Pezzo -varieties and analyze quintic curves with large stabilizers to achieve a complete classification in positive and mixed characteristics; they also study degenerations and liftability to Witt rings. Arithmetic consequences include finite Shafarevich sets for Mukai–Umemura and -types, an infinite Shafarevich set for -type, and a detailed description of good reduction phenomena, including unit-equation and Br- and cohomology-based criteria. They prove the existence of a -variety over but show that no such variety over has a generically positive-dimensional automorphism group, tying geometric classification to arithmetic constraints. These results significantly extend the understanding of genus- Fano threefolds in positive/mixed characteristic and illuminate the interplay between geometry, deformation theory, and arithmetic.

Abstract

We study prime Fano threefolds of genus 12 (-varieties) with positive-dimensional automorphism groups in positive and mixed characteristic. We classify such varieties over any perfect field. In particular, we prove that -varieties of Mukai-Umemura type over exist if and only if , . We also prove the same result for -type. As arithmetic applications, we show that the Shafarevich conjecture holds for -varieties of Mukai-Umemura type and of -type, while it fails for -varieties of -type. Moreover, we prove that there exists -varieties over , whereas there do not exist -varieties over whose generic fiber has a positive-dimensional automorphism group.
Paper Structure (25 sections, 65 theorems, 142 equations)

This paper contains 25 sections, 65 theorems, 142 equations.

Table of Contents

  1. Introduction
  2. Classification of $V_{22}$-varieties with positive-dimensional automorphism groups
  3. Arithmetic applications
  4. Methods
  5. Organization of the paper
  6. Preliminaries
  7. Basic definitions and terminologies
  8. Reviews of $V_5$-varieties
  9. Quintic curves with large stabilizers in $V_5$ varieties
  10. Characteristic $\neq 2$
  11. Characteristic $2$
  12. Degeneration of rational quintic curves with large stabilizers
  13. Two-ray games for $V_{22}$-varieties
  14. Generalities on $V_{22}$
  15. Two-ray game for conics
  16. ...and 10 more sections

Key Result

Theorem 1.1

Let $k$ be an algebraically closed field, and $X$ a $V_{22}$-variety such that $\dim \mathop{\mathrm{\mathrm{Aut}}}\nolimits_{X/k} \geq 1$. Then $\mathop{\mathrm{\mathrm{Aut}}}\nolimits_{X/k,\mathrm{red}}^{\circ}$ is one of ${\mathbb G_{m}}$, ${\mathbb G_{a}}$, and $\mathop{\mathrm{PGL}}\nolimits_{2

Theorems & Definitions (149)

  • Theorem 1.1: Theorems \ref{['thm:MukaiUmemuraclassification']}, \ref{['thm:Gaclassification']}, and \ref{['thm:Gmclassification']}
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • ...and 139 more