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On the inverse Galois problem for del Pezzo surfaces of degree 1

Luke Karras

Abstract

We solve the inverse Galois problem for del Pezzo surfaces of degree 1 over finite fields completely for 85 of the 112 possible types. We also determine for all 112 types the smallest field of existence. As an aside, we provide an example of a del Pezzo surface of degree 1 in characteristic 2 with more than one generalized Eckardt point.

On the inverse Galois problem for del Pezzo surfaces of degree 1

Abstract

We solve the inverse Galois problem for del Pezzo surfaces of degree 1 over finite fields completely for 85 of the 112 possible types. We also determine for all 112 types the smallest field of existence. As an aside, we provide an example of a del Pezzo surface of degree 1 in characteristic 2 with more than one generalized Eckardt point.

Paper Structure

This paper contains 12 sections, 34 theorems, 22 equations, 10 tables.

Table of Contents

  1. Introduction
  2. Foundations on del Pezzo surfaces
  3. Blow-up model, -1-curves, and root systems
  4. Del Pezzo surfaces over non-closed fields
  5. Anticanonical Systems
  6. The inverse Galois problem
  7. Proof structure
  8. A lower bound
  9. Blow-ups of P^2_F_q
  10. Sextics in P_F_q(1,1,2,3)
  11. Open cases
  12. Appendix

Key Result

Theorem 1.2

The following holds for the types of del Pezzo surfaces of degree 1 over finite fields: $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (61)

  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4
  • Definition 2.5
  • Definition 2.6
  • ...and 51 more