Cylinders in del Pezzo fibrations

We show that a del Pezzo fibration π: V → W of degree d contains a vertical open cylinder, that is, an open subset whose intersection with the generic fiber of π is isomorphic to Z × A_K¹ for some quasi-projective variety Z defined over the function field K of W, if and only if d ≥ 5 and π: V → W admits a rational section. We also construct twisted cylinders in total spaces of threefold del Pezzo fibrations π: V → P¹ of degree d ≤ 4.