Cylinders in del Pezzo fibrations

Abstract: We show that a del Pezzo fibration $\pi$ : V $\rightarrow$ W of degre d contains a vertical open cylinder, that is, an open subset whose intersection with the generic fiber of $\pi$ is isomorphic to $Z\times\mathbb{A}_{K}^{1}$ for some quasi-projective variety Z defined over the function field K of W , if and only if d $\ge$ 5 and $\pi$ : V $\rightarrow$ W admits a rational section. We also construct twisted cylinders in total spaces of threefold del Pezzo fibrations $\pi$ : V $\rightarrow$ P 1 of degree d $\le$ 4.