Configurations of Singular Fibers on Elliptic Surfaces

Given a list of singular fibers from the whole zoo of Kodaira fibers, is there an elliptic surface varying in a certain class (e.g. a rational, K3, Enriques surface) with this precise configuration? This problem dates back to the mid 80's, and it appears in several papers by Beauville, Miranda and Persson. While the classical theory provides many restrictions to these configurations, it lacks constructive methods to prove the existence of the possible configurations. In this talk I will briefly recall the basic general theory of elliptic surfaces, present some examples, and then show various methods to decide whether a prescribed list of fibers can exist or not in the rational and K3 cases. In particular we'll see how the problem in the rational case boils down to a combinatorial exercise, and how the K3 case is resolved by studying the effect of appropriate quotients by certain symplectic automorphisms.