Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Automorphisms of del Pezzo surfaces without points

Constantin Shramov, Anastasia Vikulova

TL;DR

The paper establishes sharp Jordan-constant bounds for automorphism groups of del Pezzo surfaces over characteristic-zero fields with no rational points, across degrees $2,4,6,8$. It combines detailed group-theoretic analyses (via Weyl groups $W(\mathrm{D}_5)$ and related configurations of $(-1)$-curves) with field-arithmetic tools (Lang–Nishimura type results and Weil restrictions) to bound $J(\operatorname{Aut}(S))$ and to exhibit sharpness through explicit constructions. The main results are: $J(\operatorname{Aut}(S))\le 168$ for $d=2$, $\le 2$ for $d=4$, $\le 4$ for $d=6$, and $\le 7200$ for $d=8$, with stronger bounds in the minimal or special-field cases (e.g., $d=8$ minimal gives $J(\operatorname{Aut}(S))\le 120$). These bounds are shown to be attainable over suitable fields, and the work highlights how the absence of $\mathbb{K}$-points imposes strong structural restrictions on automorphism groups and their birational counterparts.

Abstract

We study automorphism groups of del Pezzo surfaces without points over a field of zero characteristic, and estimate their Jordan constants.

Automorphisms of del Pezzo surfaces without points

TL;DR

The paper establishes sharp Jordan-constant bounds for automorphism groups of del Pezzo surfaces over characteristic-zero fields with no rational points, across degrees . It combines detailed group-theoretic analyses (via Weyl groups and related configurations of -curves) with field-arithmetic tools (Lang–Nishimura type results and Weil restrictions) to bound and to exhibit sharpness through explicit constructions. The main results are: for , for , for , and for , with stronger bounds in the minimal or special-field cases (e.g., minimal gives ). These bounds are shown to be attainable over suitable fields, and the work highlights how the absence of -points imposes strong structural restrictions on automorphism groups and their birational counterparts.

Abstract

We study automorphism groups of del Pezzo surfaces without points over a field of zero characteristic, and estimate their Jordan constants.
Paper Structure (6 sections, 24 theorems, 45 equations)

This paper contains 6 sections, 24 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Del Pezzo surfaces of degree $4$
  4. Del Pezzo surfaces of degree $6$
  5. Del Pezzo surfaces of degree $8$
  6. Proof of the main results

Key Result

Theorem 1.2

The Jordan constant of the birational automorphism group of the plane over an algebraically closed field of characteristic zero equals $7200$.

Theorems & Definitions (52)

  • Definition 1.1: see Popov, Popov14
  • Theorem 1.2
  • Theorem 1.3: see Shramov-SB and Shramov-SB-short
  • Remark 1.4
  • Theorem 1.5: Shramov-Cubics
  • Theorem 1.6: Vikulova-dP
  • Theorem 1.7
  • Remark 1.8
  • Proposition 1.9
  • Theorem 2.1: see e.g. VA
  • ...and 42 more