Compactifications of the affine plane over non-closed fields

TL;DR

This paper classifies certain algebraic surfaces over non-closed fields and provides criteria for the existence of affine plane cylinders in specific fibrations, advancing understanding of algebraic surface structures.

Contribution

It offers a classification of rank one del Pezzo surfaces with log canonical singularities over non-closed fields and criteria for del Pezzo fibrations containing affine plane cylinders.

Findings

Classification of normal del Pezzo surfaces over non-closed fields.

Criterion for del Pezzo fibrations with affine plane cylinders.

Insights into the structure of algebraic surfaces with singularities.

Abstract

In this article, we give the classification of normal del Pezzo surfaces of rank one with at most log canonical singularities containing the affine plane defined over an algebraically non-closed field of characteristic zero. As an application, we have a criterion for del Pezzo fibrations with canonical singularities whose generic fibers are not smooth to contain vertical $\mathbb{A}^2$-cylinders.