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Galois group of exceptional curves on the generic del Pezzo surface

Xinyu Fang

Abstract

We prove that the Galois action on the exceptional curves on the generic del Pezzo surface of degree $d$ is maximal for all degrees $d$ and over any field $k$. As a consequence of the case $d=3$, we deduce that over $\mathbb{F}_q(u)$, 100% of cubic surfaces have no Brauer-Manin obstruction.

Galois group of exceptional curves on the generic del Pezzo surface

Abstract

We prove that the Galois action on the exceptional curves on the generic del Pezzo surface of degree is maximal for all degrees and over any field . As a consequence of the case , we deduce that over , 100% of cubic surfaces have no Brauer-Manin obstruction.

Paper Structure

This paper contains 11 sections, 19 theorems, 25 equations, 1 table.

Table of Contents

  1. Introduction
  2. The generic del Pezzo surface
  3. Degrees $3\leq d\leq 7$
  4. Degrees 1 and 2
  5. Monodromy formulation and irreducibility of the incidence scheme
  6. Definition of $\mathscr{Y}_d$
  7. Irreducibility of $\mathscr{Y}_d$
  8. Monodromy formulation
  9. Proof of the main theorem
  10. Consequence for Brauer-Manin obstruction on cubic surfaces
  11. Acknowledgement

Key Result

Theorem 1.1

Let $X$ be a del Pezzo surface of degree $d$ over $k$. Then $X_{k_s}$ is isomorphic to the blow-up of $\mathbb{P}^2_{k_s}$ at $(9-d)$ points in general position in $\mathbb{P}^2(k_s)$, or $d=8$ and $X$ is isomorphic to $\mathbb{P}^1_{k_s}\times \mathbb{P}^1_{k_s}$. $\blacktriangleleft$$\blacktriangl

Theorems & Definitions (33)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Corollary 1.4
  • Lemma 2.1
  • proof
  • Remark 2.2
  • Theorem 2.3: kollar1996rational, Theorem III.3.5
  • Lemma 2.4: kollar1996rational, V.1.3.7
  • Lemma 3.1
  • ...and 23 more