Affine cones over Fano threefolds and additive group actions

TL;DR

This paper investigates when affine cones over certain Fano threefolds admit additive group actions, providing new examples of Fano threefolds with open polar cylinders that facilitate such symmetries.

Contribution

It introduces new examples of Fano threefolds of index 1 and Picard number 1 that contain open polar cylinders, linking geometric properties to additive group actions.

Findings

New examples of Fano threefolds with open polar cylinders

Criteria connecting additive group actions to geometric structures

Extension of previous classifications of Fano threefolds with cylinders

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.