Title Cylinders in Fano varieties

A cylinder in a Fano variety is an open ruled affine subset whose complement is a support of an effective anticanonical Q-divisor. This notion links together affine, birational and Kahler geometries. I prove existence and non-existence of cylinders in smooth and mildly singular (with at most du Val singularities) del Pezzo surfaces. In particular, I will answer an old question of Zaidenberg and Flenner about additive group actions on the cubic Fermat affine threefold cone. This is a joint work with Martinez-Garcia, Park and Won.