Cylinders in del Pezzo fibrations

We show that a del Pezzo fibration $π$ : V $\rightarrow$ W of degre d contains a vertical open cylinder, that is, an open subset whose intersection with the generic fiber of $π$ 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 $π$ : V $\rightarrow$ W admits a rational section. We also construct twisted cylinders in total spaces of threefold del Pezzo fibrations $π$ : V $\rightarrow$ P 1 of degree d $\le$ 4.

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