| This is a report on a joint work with Takashi Kishimoto and Yuri Prokhorov. In a paper by Hubert Flenner and the speaker, the following question was raised. Consider the affine cubic Fermat 3-fold X in the affine space of dimention 4 given by equation Does this cubic admit a non-trivial additive group action? More generally, we address the following question: Determine the affine cones over smooth projective varieties which admit an action of a unipotent algebraic group. While the original question on the Fermat cubic cone remains open, we provide a geometric criterion that relates the existence of an additive group action on the affine cone over a smooth projective variety, on one side, and the existence of an open polar cylinder in the projective variety, on the other. Due to this criterion, the affine cones over smooth del Pezzo surfaces of degree 4 possess such an action. We construct also families of Fano threefolds of index 1, Picard number 1, and of genera 9 and 10, for which the affine cone admits a unipotent group action. In fact, the automorphism group of every such cone is infinite dimensional, while the automorphism group of the underlying Fano threefold is finite. As another ingredients we explore certain Sarkisov links and a fine structure of the variety of lines on our Fano threefold. |