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Completions of the affine $3$-space into del Pezzo fibrations

Adrien Dubouloz, Takashi Kishimoto, Masaru Nagaoka

Abstract

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 $\barπ:B_{h}\to\mathbb{P}^{1}$.

Completions of the affine $3$-space into del Pezzo fibrations

Abstract

We give constructions of completions of the affine -space into total spaces of del Pezzo fibrations of every degree other than over the projective line. We show in particular that every del Pezzo surface other than blown-up in one or two points can appear as a closed fiber of a del Pezzo fibration whose total space is a -factorial threefold with terminal singularities which contains as the complement of the union of a closed fiber of and a prime divisor horizontal for . For such completions, we also give a complete description of integral curves that can appear as general fibers of the induced morphism .
Paper Structure (18 sections, 20 theorems, 20 equations)

This paper contains 18 sections, 20 theorems, 20 equations.

Table of Contents

  1. Del Pezzo fibration completions from del Pezzo surfaces pencils
  2. Pencils of divisors and their resolutions
  3. del Pezzo fiber spaces and relative MMP's
  4. Special del Pezzo pencils on $\mathbb{Q}$-Fano threefolds of divisor class rank one
  5. Applications to the construction of completions of the affine $3$-space
  6. del Pezzo fibration completions from special pencils on weighted projective spaces
  7. Examples obtained from special pencils on the projective space $\mathbb{P}^{3}$
  8. Examples obtained from special pencils on the weighted projective space $\mathbb{P}(1,1,1,2)$
  9. Examples obtained from special pencils on the weighted projective space $\mathbb{P}(1,1,2,3)$
  10. del Pezzo fibration completions from special pencils on the smooth quadric threefold in $\mathbb{P}^{4}$
  11. del Pezzo fibration completions from special pencils on the quintic del Pezzo threefold
  12. Two constructions of completions into total spaces of del Pezzo fibrations of degree $6$
  13. del Pezzo completions $(X,B_{h}\cup B_{f})$ of degree $6$ with induced rational fibrations $\bar{\pi}:B_{h}\to\mathbb{P}^{1}$
  14. del Pezzo fibrations of degree $6$ arising from conic and twisted cubic curves in the smooth quadric threefold
  15. Comparison with a construction of Prokhorov starting from the quintic del Pezzo threefold
  16. ...and 3 more sections

Key Result

Theorem 1

Let $S$ be a del Pezzo surface other than $\mathbb{P}^{2}$ blown-up in one or two points. Then there exists a completion of $\mathbb{A}^{3}$ into the total space of a del Pezzo fibration $\pi:X\to\mathbb{P}^{1}$ such that $S$ is isomorphic to a closed fiber of $\pi$ and the boundary divisor $B=X\set

Theorems & Definitions (43)

  • Theorem : Theorem A
  • Theorem : Theorem B
  • Proposition 2: DKP22
  • Definition 3
  • Lemma 4
  • Definition 5
  • Lemma 6
  • proof
  • Example 7
  • Proposition 8
  • ...and 33 more