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Automorphism groups of real rational quartic del Pezzo surfaces

Aurore Boitrel

Abstract

In this paper we give a complete description of all possible automorphism groups of real $\mathbb{R}$-rational del Pezzo surfaces $X$ of degree $4$, using the description of $X$ as the blow-up of some smooth real quadric surface $Q$ in $\mathbb{P}^{3}_{\mathbb{R}}$. We examine all possible ways to blow up $4$ geometric points on $Q$, illustrate in each case the $\operatorname{Gal}(\mathbb{C}/\mathbb{R})$-action on the conic bundle structures on $X_{\mathbb{C}}$, and use it to give a geometric description of the real automorphism group $\operatorname{Aut}_{\mathbb{R}}(X)$ by generators in terms of automorphisms and birational automorphisms of $Q$. As a consequence, we get which finite subgroups of $\operatorname{Bir}_{\mathbb{C}}(\mathbb{P}^{2})$ can act faithfully by automorphisms on real $\mathbb{R}$-rational del Pezzo surfaces of degree $4$.

Automorphism groups of real rational quartic del Pezzo surfaces

Abstract

In this paper we give a complete description of all possible automorphism groups of real -rational del Pezzo surfaces of degree , using the description of as the blow-up of some smooth real quadric surface in . We examine all possible ways to blow up geometric points on , illustrate in each case the -action on the conic bundle structures on , and use it to give a geometric description of the real automorphism group by generators in terms of automorphisms and birational automorphisms of . As a consequence, we get which finite subgroups of can act faithfully by automorphisms on real -rational del Pezzo surfaces of degree .

Paper Structure

This paper contains 12 sections, 13 theorems, 89 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries: A quick look at real quadric surfaces
  3. Automorphisms of real rational quartic del Pezzo surfaces
  4. Classical statement over the complex numbers
  5. Real automorphism groups: a geometric description of the families of automorphisms
  6. The del Pezzo surfaces obtained by blowing up $Q_{3,1}$
  7. Case 1: $X \cong Q_{3,1}(0,2)$
  8. Case 2: $X \cong Q_{3,1}(2,1)$
  9. Case 3: $X \cong Q_{3,1}(4,0)$
  10. The del Pezzo surfaces obtained by blowing up $Q_{2,2}$
  11. Case 4: $X \cong Q_{2,2}(0,2)$
  12. Case 5: $X \cong Q_{2,2}(4,0)$

Key Result

Theorem 1.1

Let $X$ be a real rational del Pezzo surface of degree $4$. Then $X$ is one of the five surfaces occuring in Table Table:Table1_summarising_main_result, where the groups $A_{0}$ and $A'$ are described in each case.

Figures (4)

  • Figure 1: Representation of the five pairs of conic bundles and the action of $\sigma$ on them.
  • Figure 2: The action of $\sigma$ on the five pairs of conic bundles.
  • Figure 3: The action of $\sigma$ on the pairs of conic bundle structures.
  • Figure 4: The action of $\mathop{\mathrm{Gal}}\nolimits(\mathbb{C}/\mathbb{R})$ on the five pairs of conic bundles.

Theorems & Definitions (31)

  • Theorem 1.1
  • Definition 2.1: Conic bundle
  • Proposition 2.2: rob16
  • Definition 3.1: bla06
  • Proposition 3.2: Structure of $\mathop{\mathrm{Aut}}\nolimits(X_{\mathbb{C}})$, bla06
  • Remark 3.3
  • Remark 3.4
  • Lemma 3.5: rob16
  • Proposition 3.6: rob16
  • proof
  • ...and 21 more