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Forms of del Pezzo surfaces of degree 5 and 6

Alexandr Zaitsev

Abstract

In this paper we obtain necessary and sufficient condition for existence of del Pezzo surfaces of degree $5$ and $6$ over a field $K$ with a prescribed action of absolute Galois group $\text{Gal} ( K^{\text{sep}}/K)$ on the graph of $(-1)$-curves. Also we compute automorphism groups of del Pezzo surfaces of degree $5$ over arbitrary fields.

Forms of del Pezzo surfaces of degree 5 and 6

Abstract

In this paper we obtain necessary and sufficient condition for existence of del Pezzo surfaces of degree and over a field with a prescribed action of absolute Galois group on the graph of -curves. Also we compute automorphism groups of del Pezzo surfaces of degree over arbitrary fields.
Paper Structure (9 sections, 21 theorems, 20 equations, 1 figure)

This paper contains 9 sections, 21 theorems, 20 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Construction of del Pezzo surfaces of degree 5. The first construction
  4. Construction of del Pezzo surfaces of degree 5. The second construction
  5. Points of the affine line and the action of a Galois group
  6. Realization of types of del Pezzo surfaces of degree 5 over infinite fields
  7. Realization of types of del Pezzo surfaces of degree 5 over finite fields
  8. Automorphism groups of del Pezzo surfaces of degree 5
  9. Classification of del Pezzo surfaces of degree 6

Key Result

Theorem 1.2

There exists a del Pezzo surface of degree $5$ of type $[H]$ over $K$ if and only if there exists a Galois extension of fields $L\supset K$ with Galois group isomorphic to $H$.

Figures (1)

  • Figure 1:

Theorems & Definitions (52)

  • Definition 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Proposition 1.6
  • Theorem 1.7
  • Remark 2.1
  • Lemma 2.2
  • Proof 1
  • ...and 42 more