Cylinders in canonical del Pezzo fibrations

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.In this article, let k be a field of characteristic zero (not necessarily an algebraically closed field) and let k be the algebraic closure of k.

Notes on cylinders in smooth projective surfaces

“…Since S is then rational over k, we know that S contains an A 1 k -cylinder by Theorem 1.7. Meanwhile, since S is a normal del Pezzo surface of the Picard rank one and of degree 1 with only two Du Val singularities of type D 4 , S does not contain any A 1 k -cylinder by [16]. Note that this fact also follow from [3] because of ρ k (S k ) = 1.…”

“…Recall that the existing condition of A 1 -cylinders in normal del Pezzo surfaces of Picard rank one with only Du Val singularities is completely determined ( [16]). By using Corollary 6.2, we can construct many examples of singular del Pezzo surfaces containing no A 1 -cylinder over algebraically non-closed fields (Picard rank of these surfaces need not necessarily be one).…”