Explicit biregular/birational geometry of affine threefolds: completions of $A^3$ into del Pezzo fibrations and Mori conic bundles - Archive ouverte HAL Accéder directement au contenu
Chapitre D'ouvrage Année : 2017

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

Résumé

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 $A^2$.
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Dates et versions

hal-01183351 , version 1 (07-08-2015)

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Adrien Dubouloz, Takashi Kishimoto. Explicit biregular/birational geometry of affine threefolds: completions of $A^3$ into del Pezzo fibrations and Mori conic bundles. Kayo Masuda; Takashi Kishimoto; Hideo Kojima; Masayoshi Miyanishi; Mikhail Zaidenberg. Algebraic varieties and automorphism groups, 75, Mathematical Society of Japan, pp.49-71, 2017, Advanced studies in pure mathematics, 978-4-86497-048-8. ⟨hal-01183351⟩
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