Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Existence of minimal del Pezzo surfaces of degree 1 with conic bundles over finite fields

Manoy T. Trip

Abstract

We study minimal del Pezzo surfaces of degree 1 with a conic bundle over a finite field $\mathbb{F}_q$ according to the action of the absolute Galois group on the singular fibers (which is known as their type). We give a lower bound on the size of the field over which they exist, and determine values of $q$ for which certain types cannot exist. In particular, we solve the inverse Galois problem for certain types of minimal del Pezzo surfaces of degree 1 over finite fields with a conic bundle structure. Additionally, we give bounds on the values of $q$ for which del Pezzo surfaces of degree 1 of index 8 exist over $\mathbb{F}_q$.

Existence of minimal del Pezzo surfaces of degree 1 with conic bundles over finite fields

Abstract

We study minimal del Pezzo surfaces of degree 1 with a conic bundle over a finite field according to the action of the absolute Galois group on the singular fibers (which is known as their type). We give a lower bound on the size of the field over which they exist, and determine values of for which certain types cannot exist. In particular, we solve the inverse Galois problem for certain types of minimal del Pezzo surfaces of degree 1 over finite fields with a conic bundle structure. Additionally, we give bounds on the values of for which del Pezzo surfaces of degree 1 of index 8 exist over .
Paper Structure (28 sections, 55 theorems, 41 equations, 4 tables)

This paper contains 28 sections, 55 theorems, 41 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Existing literature
  3. Our contribution
  4. Methods in relation to the literature
  5. Organization of the paper
  6. Background
  7. Del Pezzo surfaces
  8. Conic bundles
  9. Automorphisms of the geometric Picard group of a del Pezzo surface
  10. Automorphisms of the geometric Picard group of a standard rational conic bundle
  11. The inverse Galois problem for del Pezzo surfaces of degree 1
  12. Classification of minimal standard conic bundles of degree 1
  13. Existence of minimal rational conic bundles
  14. Description of Galois orbits of $(-1)$-curves
  15. Galois orbits of $(-1)$-curves in singular fibers of conic bundle
  16. ...and 13 more sections

Key Result

Theorem 1.1

Theorems & Definitions (114)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.1: iskovskih
  • Definition 2.2
  • Theorem 2.3: manin
  • Theorem 2.4: dolgachev
  • Definition 2.5
  • Remark 2.6
  • Definition 2.7
  • ...and 104 more