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

We study certain pencils (f) over bar : F -> 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 P and an appropriate multiple of a hyperplane H. Our main observation is that every minimal model program relative to the morphism (f) over tilde : (P) over tilde -> P-1 lifting (f) over bar on a suitable resolution sigma : (P) over tilde -> P of its indeterminacies pre- serves the open subset sigma(-1)(P\textbackslashH)similar or equal to A(3). As an application, we obtain projective completions of A(3) into del Pezzo fibrations over P-1 of every degree less than or equal to 4. We also obtain completions of A(3) into Mori conic bundles, whose restrictions to A(3) are twisted A(1)*-fibrations over A(2).