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 $\bar{\pi }:B_h\to {\mathbb{P}}^1$.

16 pages