  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.
