For a projective variety X and an ample divisor H on it, an H-polar cylinder in X is an open ruled affine subset whose complement is a support of an effective Q-divisor Q-rationally equivalent to H. This notion links together affine, birational and Kahler geometries. I will show how to prove existence and non-existence of H-polar cylinders in smooth and mildly singular del Pezzo surfaces (for different polarizations). The obstructions comes from log canonical thresholds and Fujita numbers. As an application, 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 Jihun Park (POSTECH) and Joonyeong Won (KAIST).