Completions of the affine 3-space into del Pezzo fibrations

Abstract

We give constructions of completions of the affine 3-space into total spaces of del Pezzo fibrations of every degree other than 7 over the projective line. We show in particular that every del Pezzo surface other than \({\mathbb {P}}^{2}\) blown-up in one or two points can appear as a closed fiber of a del Pezzo fibration \(\pi :X\rightarrow {\mathbb {P}}^{1}\) whose total space X is a \({\mathbb {Q}}\)-factorial threefold with terminal singularities which contains \({\mathbb {A}}^{3}\) as the complement of the union of a closed fiber of \(\pi \) and a prime divisor \(B_{h}\) horizontal for \(\pi \). For such completions, we also give a complete description of integral curves that can appear as general fibers of the induced morphism \(\bar{\pi }:B_{h}\rightarrow {\mathbb {P}}^{1}\).