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The largest automorphism group of del Pezzo surface of degree $2$ without points

Anastasia V. Vikulova

TL;DR

This paper exhibits a field and a del Pezzo surface of degree $2$ with no rational points yet achieving the largest possible automorphism group over an algebraically closed field of characteristic zero. The construction uses a weighted projective model in $\mathbb{P}(2,1,1,1)$ defined over $\mathbb{Q}(\sqrt{-7})$ and the associated double cover branched along a quartic curve $C$, whose automorphism group is $\mathrm{PSL}_2(\mathbb{F}_7)$. The automorphism group of the surface is $\mathrm{PSL}_2(\mathbb{F}_7) \times \mathbb{Z}/2\mathbb{Z}$, where the extra $\mathbb{Z}/2\mathbb{Z}$ comes from the Geiser involution; the Picard group is $\mathrm{Pic}(S) \cong \mathbb{Z}$. A key ingredient is a $2$-adic obstruction at the place above $2$ showing that $S(\mathbb{Q}(\sqrt{-7}))=\emptyset$, realized via a congruence modulo $64$ for a defining form. The work demonstrates that arithmetic limitations can prevent points while still allowing the maximal symmetry group to be realized.

Abstract

We construct an example of a field and a del Pezzo surface of degree $2$ over this field without points such that its automorphism group is isomorphic to $\mathrm{PSL}_2(\mathbb{F}_7) \times \mathbb{Z}/2\mathbb{Z},$ which is the largest possible automorphism group of del Pezzo surface of degree $2$ over an algebraically closed field of characteristic zero.

The largest automorphism group of del Pezzo surface of degree $2$ without points

TL;DR

This paper exhibits a field and a del Pezzo surface of degree with no rational points yet achieving the largest possible automorphism group over an algebraically closed field of characteristic zero. The construction uses a weighted projective model in defined over and the associated double cover branched along a quartic curve , whose automorphism group is . The automorphism group of the surface is , where the extra comes from the Geiser involution; the Picard group is . A key ingredient is a -adic obstruction at the place above showing that , realized via a congruence modulo for a defining form. The work demonstrates that arithmetic limitations can prevent points while still allowing the maximal symmetry group to be realized.

Abstract

We construct an example of a field and a del Pezzo surface of degree over this field without points such that its automorphism group is isomorphic to which is the largest possible automorphism group of del Pezzo surface of degree over an algebraically closed field of characteristic zero.

Paper Structure

This paper contains 3 sections, 10 theorems, 43 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['th:nopoints']}

Key Result

Theorem 1.1

Let $G$ be a finite group. If there is a field $\mathbf{F}$ of characteristic zero and a non-trivial Severi--Brauer surface $S$ over $\mathbf{F}$ such that the group $\mathrm{Aut}(S)$ contains a subgroup isomorphic to $G,$ then there is $n \in \mathbb{N}$ divisible only by primes congruent to $1$ mo

Theorems & Definitions (17)

  • Theorem 1.1: cf. ShramovfgSV
  • Theorem 1.2: VikulovaSB
  • Theorem 1.3: Shramovcubic
  • Theorem 1.4
  • Lemma 2.1: Serrearithmetic
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Remark 2.4
  • ...and 7 more