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 $\overline{f}:\mathbb{P}\dashrightarrow\mathbb{P}^{1}$ of del Pezzo surfaces generated by a smooth del Pezzo surface $S$ of degree less than or equal to 3 anti-canonically embedded into a weighted projective space $\mathbb{P}$ and an appropriate multiple of a hyperplane $H$. Our main observation is that every minimal model program relative to the morphism $\tilde{f}:\tilde{\mathbb{P}}\rightarrow\mathbb{P}^{1}$ lifting $\overline{f}$ on a suitable resolution $\sigma:\tilde{\mathbb{P}}\rightarrow\mathbb{P}$ of its indeterminacies preserves the open subset $\sigma^{-1}(\mathbb{P}\setminus H)\simeq\mathbb{A}^{3}$. As an application, we obtain projective completions of $\mathbb{A}^{3}$ into del Pezzo fibrations over $\mathbb{P}^{1}$ of every degree less than or equal to 4. We also obtain completions of $\mathbb{A}^{3}$ into Mori conic bundles, whose restrictions to $\mathbb{A}^{3}$ are twisted $\mathbb{A}_{*}^{1}$-fibrations over $\mathbb{A}^{2}$.