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We study certain pencils of del Pezzo surfaces generated by a smooth delPezzo surface S of degree less or equal to 3 anti-canonically embedded into aweighted projective space P and an appropriate multiple of a hyperplane H. Ourmain observation is that every minimal model program relative to the morphismlifting such pencil on a suitable resolution of its indeterminacies preservesthe open subset P H \^a A^3. As an application, we obtain projectivecompletions of A^3 into del Pezzo fibrations over P^1 of every degree less orequal to 4. We also obtain completions of A^3 into Mori conic bundles, whoserestrictions to A^3 are twisted C*-fibrations over A^2 .

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