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

TL;DR

This paper explores the explicit construction of affine threefolds as completions into del Pezzo fibrations and Mori conic bundles, revealing new geometric structures and classifications.

Contribution

It provides explicit methods to complete A^3 into del Pezzo fibrations and Mori conic bundles, expanding understanding of affine threefold geometry.

Findings

Constructed projective completions of A^3 into del Pezzo fibrations of degrees ≤ 4.

Established preservation of the affine part during minimal model programs.

Developed completions of A^3 into Mori conic bundles with twisted C*-fibrations.

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 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 .