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 $\pi:X\to\mathbb{P}^{1}$ 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 $\bar{\pi}:B_{h}\to\mathbb{P}^{1}$.