Cylinders in Fano threefolds

A variety $X$ is called cylindrical if it contains a Zariski-open subset isomorphic to $Z \times \mathbb A^1$ for some affine variety $Z$. If such an open subset is the complement of the support of an effective $\mathbb{Q}$-divisor $D$ satisfying $D \sim_{\mathbb Q} -K_X$, then it is called an anticanonical polar cylinder. The study of cylinders in Fano varieties lies at the intersection of birational geometry and affine geometry. In particular, over an algebraically closed field of characteristic zero, a cylindrical Fano threefold is rational. In recent years, many results have been obtained on which Fano varieties admit cylinders and which do not. In this talk, I will discuss several results on the existence and non-existence of cylinders in Fano threefolds, together with some of the main geometric methods used to prove them.