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

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 $P \ H ≃ A^3$. 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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Type de document :
Chapitre d'ouvrage
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. 〈http://mathsoc.jp/publication/ASPM/aspmlist.html〉

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https://hal.archives-ouvertes.fr/hal-01183351
Soumis le : vendredi 7 août 2015 - 12:29:46
Dernière modification le : vendredi 8 juin 2018 - 14:50:07
Document(s) archivé(s) le : mercredi 26 avril 2017 - 10:07:55

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DelPezzo-Completions.pdf
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• HAL Id : hal-01183351, version 1
• ARXIV : 1508.01792

### Citation

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. 〈http://mathsoc.jp/publication/ASPM/aspmlist.html〉. 〈hal-01183351〉

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