[Paper Review] New examples of cylindrical Fano fourfolds

Yuri Prokhorov, Mikhail Zaidenberg|arXiv (Cornell University)|Jul 7, 2015

Algebraic Geometry and Number Theory21 references4 citations

TL;DR

This paper constructs new families of smooth Fano fourfolds with Picard rank 1 that contain cylinders—Zariski open subsets isomorphic to $ Z \times \mathbb{A}^1 $—via Sarkisov links starting from a del Pezzo fourfold of degree 5. The key contribution is the explicit construction of cylindrical Mukai fourfolds of genus 9 and 10, which implies their affine cones admit effective $ \mathbb{G}_a $-actions.

ABSTRACT

We construct new families of smooth Fano fourfolds with Picard rank 1, which contain cylinders, i.e., Zariski open subsets of form $Z imes A^1$, where $Z$ is a quasiprojective variety. The affine cones over such a fourfold admit effective $G_a$-actions. Similar constructions of cylindrical Fano threefolds and fourfolds were done previously in [KPZ11, KPZ14, PZ15].

Motivation & Objective

• To construct new families of smooth Fano fourfolds with Picard rank 1 that contain cylinders.
• To extend previous results on cylindrical Fano threefolds and fourfolds to higher genus Mukai fourfolds.
• To establish the existence of cylindrical Mukai fourfolds of genus 9 and 10 using geometric links from a quintic del Pezzo fourfold.
• To provide evidence for the existence of cylindrical Mukai fourfolds in higher genera, addressing open questions on cylindricity and rationality.

Proposed method

• Utilize Sarkisov links starting from a smooth del Pezzo fourfold $ W_5 \subset \mathbb{P}^7 $ of degree 5.
• Construct the links via blowups along specific surfaces: a rational normal quintic scroll $ F \cong \mathbb{F}_1 $ or an anticanonical del Pezzo surface of degree 6.
• Define the forward map $ \phi: W \dashrightarrow V \subset \mathbb{P}^{g+2} $ as the linear system of quadrics passing through $ F $, and the inverse via projection from the linear span of a surface $ S \subset V $.
• Verify that the resulting $ V $ is a smooth Mukai fourfold of genus $ g=9 $ or $ g=10 $, with $ \operatorname{Pic}(V) = \mathbb{Z} \cdot L $ and $ -K_V = 2L $.
• Use the isomorphism $ V \setminus \varphi(D) \cong W \setminus \rho(\tilde{E}) $ to show that the image contains a cylinder, inherited from the original $ W $.
• Apply known criteria from [KPZ13] to conclude that the affine cone over $ V $ admits an effective $ \mathbb{G}_a $-action if and only if $ V $ is cylindrical.

Experimental results

Research questions

• RQ1 Can new families of cylindrical Mukai fourfolds of genus 9 and 10 be constructed via geometric links from a del Pezzo fourfold of degree 5?
• RQ2 What conditions on embedded surfaces in $ W_5 $ ensure the existence of Sarkisov links leading to smooth cylindrical Mukai fourfolds?
• RQ3 Do all Mukai fourfolds of genus $ g \geq 7 $ admit cylindrical structures, or are such examples sparse?
• RQ4 How does the existence of a cylinder in a Fano fourfold relate to rationality and compactifications of $ \mathbb{A}^4 $?

Key findings

• The paper constructs two new families of smooth cylindrical Mukai fourfolds: $ V_{16} \subset \mathbb{P}^{11} $ of genus 9 and $ V_{18} \subset \mathbb{P}^{12} $ of genus 10.
• These fourfolds arise via Sarkisov links from a del Pezzo fourfold $ W_5 \subset \mathbb{P}^7 $ of degree 5, using specific surfaces $ F \subset W_5 $ of type $ \mathbb{F}_1 $ or anticanonical sextic del Pezzo.
• The existence of a cylinder in $ V $ is established via the isomorphism $ V \setminus \varphi(D) \cong W \setminus \rho(\tilde{E}) $, inherited from the original $ W $.
• The affine cone over each such $ V $ admits an effective $ \mathbb{G}_a $-action, as guaranteed by the criterion in [KPZ13].
• For $ g=9 $ and $ g=10 $, the Mukai fourfolds are rational, as shown via the Fano-Iskovskikh double projection and Tsen’s theorem over $ \mathbb{C}(\mathbb{P}^1) $.
• The constructions confirm that cylindrical Mukai fourfolds exist in genera 9 and 10, extending previous results for $ g=7,8 $.