Completions of the affine 3-space into del Pezzo fibrations (Q6592700)

The article concerns very special completion of the affine 3-space \(\mathbb{A}^{3}\) over an algebraically closed field of characteristic zero, connected with del Pezzo surfaces (they are nonsingular algebraic surfaces \(X\) for which anticanonical divisor \( -K_{X}\) is ample). The main theorem says that for any del Pezzo surface \(S\) other than \(\mathbb{P}^{2}\), blown-up in one or two points, there exists a completion of \(\mathbb{A}^{3}\) into the total space of a del Pezzo fibration \(\pi :X \rightarrow \mathbb{P}^{1}\) (= the general fiber is a del Pezzo surface) such that \(S\) is isomorphic to a closed fiber of \(\pi \) and the boundary divisor \(B =X \setminus \mathbb{A}^{3}\) is the union of a fiber \(B_{f}\) of \(\pi \) and a prime divisor \(B_{h}\) which dominates \(\mathbb{P}^{1} .\)