Affine cones over Fano threefolds and additive group actions

Takashi Kishimoto (1) , Yuri Prokhorov (2) , Mikhail Zaidenberg (3)

1 Department of Mathematics, Faculty of Science, Saitama University
2 SMI | RAS - Steklov Mathematical Institute [Moscow]
3 IF - Institut Fourier

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.

Mots clés: Additive group; Group action; Affine cone; Fano variety; Automorphism

Domaines: Géométrie algébrique [math.AG]

Takashi Kishimoto, Yuri Prokhorov, Mikhail Zaidenberg. Affine cones over Fano threefolds and additive group actions. Osaka Journal of Mathematics, 2014, 51 (4), pp.1093-1112. ⟨hal-00598569⟩