Group actions on affine cones

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 ≥ 4 possess such an action. A question in [FZ1] 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.