Cylindricity of weighted singular del Pezzo surfaces over fields of characteristic zero

TL;DR

This paper investigates the cylindricity property of certain singular del Pezzo surfaces derived from weighted projective planes over fields of characteristic zero, with applications to higher-dimensional fibrations.

Contribution

It characterizes the cylindricity of $Bbbk$-forms of singular del Pezzo surfaces obtained via blow-ups, extending understanding of their geometric structure.

Findings

Identifies conditions under which these surfaces are cylindrical.

Constructs vertical cylinders on higher-dimensional fibrations with such fibers.

Provides new insights into the geometry of del Pezzo surfaces over arbitrary fields.

Abstract

In this paper, we study the cylindricity of $\Bbbk$-forms of singular del Pezzo surfaces obtained by blowing up weighted projective planes $\mathbb{P}(1,1,m)$ over an arbitrary field $\Bbbk$ of characteristic zero. As an application, we obtain vertical cylinders on higher-dimensional fibrations whose generic fibers are such $\Bbbk$-forms.