[Paper Review] Cylinders in del Pezzo fibrations

Adrien Dubouloz, Takashi Kishimoto|arXiv (Cornell University)|Jul 4, 2016

Algebraic Geometry and Number Theory8 references3 citations

TL;DR

This paper establishes a complete characterization of del Pezzo fibrations admitting vertical cylinders: a del Pezzo fibration $\pi: V \to W$ of degree $d$ contains a vertical $\mathbb{A}^1$-cylinder if and only if $d \geq 5$ and $\pi$ admits a rational section. For fibrations of degree $d \leq 4$, the authors construct twisted cylinders in the total space, showing that such fibrations can still contain affine threefolds as open subsets despite lacking vertical cylinders.

ABSTRACT

We show that a del Pezzo fibration $π$ : V $ ightarrow$ W of degre d contains a vertical open cylinder, that is, an open subset whose intersection with the generic fiber of $π$ is isomorphic to $Z imes\mathbb{A}_{K}^{1}$ for some quasi-projective variety Z defined over the function field K of W , if and only if d $\ge$ 5 and $π$ : V $ ightarrow$ W admits a rational section. We also construct twisted cylinders in total spaces of threefold del Pezzo fibrations $π$ : V $ ightarrow$ P 1 of degree d $\le$ 4.

Motivation & Objective

• To characterize when a del Pezzo fibration $\pi: V \to W$ contains a vertical $\mathbb{A}^1$-cylinder, i.e., an open subset isomorphic to $Z \times \mathbb{A}^1$ over the function field of $W$.
• To investigate the existence of twisted $\mathbb{A}^1$-cylinders in total spaces of del Pezzo fibrations of degree $d \leq 4$, which are not vertical but still contain affine threefolds.
• To provide a complete classification of cylinders in del Pezzo fibrations, extending the understanding of unipotent group actions and birational geometry in higher dimensions.
• To construct explicit families of del Pezzo fibrations $\pi: V \to \mathbb{P}^1$ of degree $d \leq 4$ whose total spaces contain twisted cylinders isomorphic to $\mathbb{A}^3$.

Proposed method

• Use of relative minimal model program (MMP) techniques over the base $W$, particularly focusing on relative contractions and extremal rays in the relative cone of curves $\overline{NE}(\tilde{\mathbb{P}}/\mathbb{P}^1)$.
• Construction of a good resolution $\sigma: \tilde{\mathbb{P}} \to \mathbb{P}^3$ of a rational map $\overline{f}: \mathbb{P}^3 \dashrightarrow \mathbb{P}^1$ via a sequence of blow-ups at points and curves.
• Computation of normal bundles and adjunction formulas to verify that certain exceptional divisors $E_6$ are isomorphic to $\mathbb{P}^1 \times \mathbb{P}^1$ with normal bundle $\mathcal{O}(-1,-1)$, ensuring they are extremal and contractible.
• Verification that the restriction $\sigma_6|_{E_6}: E_6 \to L_5$ is a trivial $\mathbb{P}^1$-bundle, which allows the identification of an extremal ray in the relative cone of curves.
• Application of the relative MMP to contract the divisor $E_6$, resulting in a new threefold where the fiber structure is preserved and the cylinder structure becomes visible.
• Use of the fact that the existence of a vertical cylinder is equivalent to the existence of an $\mathbb{A}^1$-cylinder in the generic fiber over the function field $K(W)$, reducing the problem to a fiberwise condition.

Experimental results

Research questions

• RQ1 When does a del Pezzo fibration $\pi: V \to W$ admit a vertical $\mathbb{A}^1$-cylinder, i.e., an open subset isomorphic to $Z \times \mathbb{A}^1$ for some variety $Z$ over $K(W)$?
• RQ2 What are the obstructions to the existence of vertical cylinders in del Pezzo fibrations of degree $d \leq 4$?
• RQ3 Can twisted $\mathbb{A}^1$-cylinders exist in the total space of a del Pezzo fibration of degree $d \leq 4$, even when vertical cylinders do not?
• RQ4 How can one explicitly construct del Pezzo fibrations of degree $d \leq 4$ whose total spaces contain open subsets isomorphic to $\mathbb{A}^3$?
• RQ5 What role does the existence of a rational section play in the existence of vertical cylinders in del Pezzo fibrations?

Key findings

• A del Pezzo fibration $\pi: V \to W$ admits a vertical $\mathbb{A}^1$-cylinder if and only if its degree $d \geq 5$ and $\pi$ admits a rational section.
• For del Pezzo fibrations of degree $d \leq 4$ over $\mathbb{P}^1$, no vertical $\mathbb{A}^1$-cylinders exist, due to the non-rationality of the total space and the rationality obstruction from unirationality.
• Despite the absence of vertical cylinders, the authors construct explicit families of del Pezzo fibrations $\pi: V \to \mathbb{P}^1$ of degree $d \leq 4$ whose total spaces contain twisted $\mathbb{A}^1$-cylinders isomorphic to $\mathbb{A}^3$.
• The construction relies on a good resolution of a rational map $\mathbb{P}^3 \dashrightarrow \mathbb{P}^1$ via six blow-ups, culminating in a divisor $E_6 \simeq \mathbb{P}^1 \times \mathbb{P}^1$ with normal bundle $\mathcal{O}(-1,-1)$.
• The divisor $E_6$ is shown to be extremal in the relative cone of curves, allowing its contraction in a relative MMP, which leads to the appearance of a twisted cylinder in the total space.
• The final construction yields a birational model where the generic fiber contains an $\mathbb{A}^1$-cylinder over the function field, confirming the existence of a twisted $\mathbb{A}^3$-open subset in the total space.