Cylinders in weak del Pezzo fibrations

TL;DR

This paper investigates the existence and classification of vertical cylinders in weak del Pezzo fibrations, extending previous work on del Pezzo fibrations and focusing on the geometric properties of minimal weak del Pezzo surfaces.

Contribution

It provides a classification of minimal weak del Pezzo surfaces over characteristic zero fields and shows that weak del Pezzo fibrations with vertical cylinders are highly restricted.

Findings

Classification of minimal weak del Pezzo surfaces over characteristic zero fields

Weak del Pezzo fibrations with vertical cylinders are quite restrictive

Extension of previous results on del Pezzo fibrations to weak del Pezzo cases

Abstract

In this article, we shall look into the existence of vertical cylinders contained in a weak del Pezzo fibration as a generalization of the former work due to Dubouloz and Kishimoto in which they observed that of vertical cylinders found in del Pezzo fibrations. The essence lying in the existence of a cylinder in the generic fiber, we devote mainly ourselves into a geometry of minimal weak del Pezzo surfaces defined over a field of characteristic zero from the point of view of cylinders. As a result, we give the classification of minimal weak del Pezzo surfaces defined over a field of characteristic zero, moreover, we show that weak del Pezzo fibrations containing vertical cylinders are quite restrictive.