Transversality of sections on elliptic surfaces
I will discuss some subset of the following joint work with Giancarlo Urzúa Given an elliptic fibration E->C over a field k, and with zero section O and another section P of infinite order, it’s natural to ask whether the intersections between O and multiples nP of P are transverse or whether there are tangencies between O and nP. (The number of intersections is asymptotic to a constant times n^2.). The question can be rephrased naturally in terms of “unlikely intersections” and in terms of “elliptic divisibility sequences”. We show that if k has characteristic zero, the number of tangencies is always finite. For k of characteristic not 2 or 3, we show that for “very general” data (E,P), there are no tangencies. This has a nice application to geography of surfaces. Recently we have also given a remarkable upper bound on the number of tangencies in characteristic zero.