Group actions on affine cones

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: Determine the affine cones over smooth projective varieties which admit an action of a connected algebraic group different from the standard C*-action by scalar matrices and its inverse action. We show in particular that the affine cones over anticanonically embedded smooth del Pezzo surfaces of degree at least 4 possess such an action. A question by Flenner and the third author whether this is also true for cubic surfaces, occurs to be out of reach for our methods. Nevertheless, we provide a general geometric criterion that could be helpful also in this case.

Mots clés: group action; affine cone; del Pezzo surface; automorphism group

Domaines: Géométrie algébrique [math.AG]

Takashi Kishimoto, Yuri Prokhorov, Mikhail Zaidenberg. Group actions on affine cones. Transformation Groups, 2013, 18 (4), pp.1137-1153. ⟨10.1007/s00031-013-9246-5⟩. ⟨hal-00389309v2⟩