Inverse curve problems on del Pezzo surfaces

TL;DR

This paper classifies rational lines and conic fibrations on del Pezzo surfaces over various fields, solving inverse Galois problems and generalizing classical results on cubic surfaces.

Contribution

It provides a comprehensive classification of rational curves on del Pezzo surfaces and extends inverse Galois problem solutions to new settings, including characteristic 2.

Findings

Complete classification over all degrees and fields

New solutions for inverse Galois problem in characteristic 2

Generalization of classical theorems on cubic surfaces

Abstract

We classify the number of $k$-rational lines and conic fibrations on del Pezzo surfaces over a field $k$ in terms of relatively minimal surfaces and establish rational curve analogues of the inverse Galois problem for del Pezzo surfaces. We completely solve these problems in all degrees over all global, local and finite fields and provide new solutions of the inverse Galois problem in characteristic 2. Our results generalise well-known theorems on cubic surfaces.