  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.
