Cylinders in canonical del Pezzo fibrations

TL;DR

This paper investigates the conditions under which canonical del Pezzo fibrations contain vertical cylinders, extending previous results to include cases with singularities by analyzing the degree and type of singularities on the generic fiber.

Contribution

It generalizes the criteria for the existence of vertical cylinders in del Pezzo fibrations to include those with canonical singularities, based on singularity analysis.

Findings

Criteria for vertical cylinders in singular del Pezzo fibrations

Relation between singularities and cylinder existence

Extension of previous smooth case results

Abstract

Cylinders in projective varieties play an important role in connection with unipotent group actions on certain affine algebraic varieties. The previous work due to Dubouloz and Kishimoto deals with the condition for a del Pezzo fibration to contain a vertical cylinder. In the present work, as a generalization in the sense of singularities, we shall determine the condition under which a del Pezzo fibration with canonical singularities admits a vertical cylinder by means of degree and type of singularities found on the corresponding generic fiber.