The Brill-Segre formula and the abc conjecture

This is a write-up of lectures presented at the first Arizona Winter School in Arithmetic Geometry on the abc conjecture. There isn’t anything new in these notes, except perhaps the point of view. Most of the results are in [V] and [TV]. The Brill-Segre formula counts the number of osculation points for a morphism of a curve to n-dimensional space and generalizes the Hurwitz formula (n = 1) and the Plucker formula (n = 2). The Brill-Segre formula implies the abc theorem for function fields for arbitrarily many summands. Smirnov has suggested a conjectural analogue of Hurwitz formula for number fields which implies the abc conjecture. We had hoped to be able to formulate a corresponding number field analogue of the Brill-Segre formula, but had to stop short of that goal and discuss only local aspects of such an analogue. Let X be an irreducible, nonsingular, projective algebraic curve of genus g defined over an algebraically closed field k of characteristic zero (see the papers of J. Wang [W1,2] for the case of positive characteristic). Let K be the function field of X. For elements f0, . . . , fn of K, not all zero, we define the height as