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Transversality of sections on elliptic surfaces

Algebraic Geometry Seminar

Transversality of sections on elliptic surfaces
Series: Algebraic Geometry Seminar
Location: ENR2 S395
Presenter: Doug Ulmer, University of Arizona

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.