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

TL;DR

This paper investigates the existence and classification of minimal degree 1 del Pezzo surfaces with conic bundle structures over finite fields, providing bounds on field sizes and solving specific inverse Galois problems.

Contribution

It establishes bounds on the finite fields where such surfaces exist and solves the inverse Galois problem for certain types over finite fields.

Findings

Lower bounds on field size for existence of these surfaces

Identification of field sizes where certain types cannot exist

Solutions to inverse Galois problems for specific surface types

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 $\mathbb{F}_q$.