Completions of the affine $3$-space into del Pezzo fibrations
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866913187009396736 |
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| author | Dubouloz, Adrien Kishimoto, Takashi Nagaoka, Masaru |
| author_facet | Dubouloz, Adrien Kishimoto, Takashi Nagaoka, Masaru |
| contents | We give constructions of completions of the affine $3$-space into total spaces of del Pezzo fibrations of every degree other than $7$ over the projective line. We show in particular that every del Pezzo surface other than $\mathbb{P}^{2}$ blown-up in one or two points can appear as a closed fiber of a del Pezzo fibration $π:X\to\mathbb{P}^{1}$ whose total space $X$ is a $\mathbb{Q}$-factorial threefold with terminal singularities which contains $\mathbb{A}^{3}$ as the complement of the union of a closed fiber of $π$ and a prime divisor $B_{h}$ horizontal for $π$. For such completions, we also give a complete description of integral curves that can appear as general fibers of the induced morphism $\barπ:B_{h}\to\mathbb{P}^{1}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_02857 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Completions of the affine $3$-space into del Pezzo fibrations Dubouloz, Adrien Kishimoto, Takashi Nagaoka, Masaru Algebraic Geometry 14R10, 14E30, 14J30 We give constructions of completions of the affine $3$-space into total spaces of del Pezzo fibrations of every degree other than $7$ over the projective line. We show in particular that every del Pezzo surface other than $\mathbb{P}^{2}$ blown-up in one or two points can appear as a closed fiber of a del Pezzo fibration $π:X\to\mathbb{P}^{1}$ whose total space $X$ is a $\mathbb{Q}$-factorial threefold with terminal singularities which contains $\mathbb{A}^{3}$ as the complement of the union of a closed fiber of $π$ and a prime divisor $B_{h}$ horizontal for $π$. For such completions, we also give a complete description of integral curves that can appear as general fibers of the induced morphism $\barπ:B_{h}\to\mathbb{P}^{1}$. |
| title | Completions of the affine $3$-space into del Pezzo fibrations |
| topic | Algebraic Geometry 14R10, 14E30, 14J30 |
| url | https://arxiv.org/abs/2401.02857 |