[Paper Review] Affine cones over Fano threefolds and additive group actions

Takashi Kishimoto, Yuri Prokhorov+1 more|arXiv (Cornell University)|Jun 7, 2011

Algebraic Geometry and Number Theory20 references21 citations

TL;DR

This paper establishes a geometric criterion linking the existence of an effective additive group action on the affine cone over a Fano threefold to the presence of a cylinder (an open subset isomorphic to a product with the affine line) in the threefold itself. For Fano threefolds of index 1, Picard number 1, and genus 9 or 10, the authors prove that if the Fano scheme of lines is singular, then the affine cone admits a nontrivial ${\mathbb{G}}_a$-action, providing new families of such threefolds beyond previously known cases.

ABSTRACT

We address the following question: When an affine cone over a smooth Fano threefold admits an effective action of the additive group? In this paper we deal with Fano threefolds of index 1 and Picard number 1. Our approach is based on a geometric criterion from our previous paper, which relates the existence of an additive group action on the cone over a smooth projective variety X with the existence of an open polar cylinder in X. Non-trivial families of Fano threefolds carrying a cylinder were found in loc. cit. Here we provide new such examples.

Motivation & Objective

• To determine when the affine cone over a smooth Fano threefold admits an effective action of the additive group ${\mathbb{G}}_a$.
• To extend known families of Fano threefolds with ${\mathbb{G}}_a$-actions on their affine cones beyond previously identified cases.
• To establish a geometric criterion linking the existence of a cylinder in the Fano threefold to the existence of a ${\mathbb{G}}_a$-action on its affine cone.
• To investigate the role of the Fano scheme of lines ($\tau(X)$) in determining the existence of such actions.

Proposed method

• The authors apply a geometric criterion from [KPZ] that relates the existence of a ${\mathbb{G}}_a$-action on the affine cone to the existence of a polar ${\mathbb{A}}^1$-cylinder in the Fano threefold.
• They analyze the Fano scheme $\tau(X)$ of lines on the threefold $X$, focusing on whether it is smooth or not.
• For Fano threefolds of genus 9 and 10 with Picard number 1 and index 1, they show that if $\tau(X)$ is singular, then $X$ contains a cylinder.
• The construction of cylinders relies on geometric techniques involving blow-ups and the resolution of singularities on cubic surfaces.
• They use the adjunction formula and the theory of differents to rule out the existence of certain curves on singular cubic surfaces.
• The proof involves analyzing the intersection theory on minimal resolutions of singular cubic surfaces to derive contradictions when assuming the existence of a genus 3 curve of degree 7.

Experimental results

Research questions

• RQ1 Under what conditions does the affine cone over a Fano threefold admit an effective ${\mathbb{G}}_a$-action?
• RQ2 How does the singular locus of the Fano scheme $\tau(X)$ of lines on a Fano threefold relate to the existence of a cylinder in $X$?
• RQ3 Can new families of Fano threefolds with ${\mathbb{G}}_a$-actions on their affine cones be constructed beyond previously known examples?
• RQ4 What are the obstructions to the existence of a genus 3 curve of degree 7 on a cubic surface with an $A_3$, conic, or $E_6$ singularity?

Key findings

• The affine cone over a Fano threefold $X$ of genus 9 or 10 with $\operatorname{Pic}(X) \simeq \mathbb{Z} \cdot (-K_X)$ admits an effective ${\mathbb{G}}_a$-action if the Fano scheme $\tau(X)$ is not smooth.
• The set of such Fano threefolds forms a codimension one subvariety in the moduli space of all such threefolds.
• The existence of a cylinder in $X$ is equivalent to the existence of a ${\mathbb{G}}_a$-action on the affine cone over $X$, under the Picard number one assumption.
• For cubic surfaces with $A_3$-singularities, a smooth genus 3 curve of degree 7 can exist and lead to a cylinder in the corresponding Fano threefold.
• No such curve exists on a cubic surface with an isolated conic singularity or an $E_6$-singularity, which obstructs the construction of cylinders in those cases.
• The contradiction in the differents computation shows that a genus 3 curve of degree 7 cannot coexist with a line through the $E_6$-singularity in a way consistent with the adjunction formula.