Explicit biregular/birational geometry of affine threefolds: completions of $\mathbb{A}^3$ into del Pezzo fibrations and Mori conic bundles

Abstract

We study certain pencils of del Pezzo surfaces generated by a smooth del Pezzo surface $S$ of degree less or equal to 3 anti-canonically embedded into a weighted projective space P and an appropriate multiple of a hyperplane $H$. Our main observation is that every minimal model program relative to the morphism lifting such pencil on a suitable resolution of its indeterminacies preserves the open subset $P H ≃ A^3$. As an application, we obtain projective completions of $A^3$ into del Pezzo fibrations over $P^1$ of every degree less or equal to 4. We also obtain completions of $A^3$ into Mori conic bundles, whose restrictions to $A^3$ are twisted $C*$-fibrations over $A^2$.