Completions of the affine 3-space into del Pezzo fibrations

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 \({\mathbb {P}}^{2}\) ${\mathbb{P}}^2$ blown-up in one or two points can appear as a closed fiber of a del Pezzo fibration \(\pi :X\rightarrow {\mathbb {P}}^{1}\) $\pi :X\to {\mathbb{P}}^1$ whose total space X is a \({\mathbb {Q}}\) $\mathbb{Q}$ -factorial threefold with terminal singularities which contains \({\mathbb {A}}^{3}\) ${\mathbb{A}}^3$ as the complement of the union of a closed fiber of \(\pi \) $\pi$ and a prime divisor \(B_{h}\) $B_h$ horizontal for \(\pi \) $\pi$ . For such completions, we also give a complete description of integral curves that can appear as general fibers of the induced morphism \(\bar{\pi }:B_{h}\rightarrow {\mathbb {P}}^{1}\) $\overline{\pi }:B_h\to {\mathbb{P}}^1$ .