Arizona Winter School 1998
Voloch's Course Abstract

The Brill-Segre formula and the abc conjecture

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 generalization of the abc theorem for function fields (due to Mason) to arbitrarily many summands (proved by Brownawell, Masser and Voloch). Smirnov has suggested a conjectural analogue of Hurwitz formula for number fields which implies the abc conjecture. We hope to be able to formulate a corresponding number field analogue of the Brill-Segre formula. The talks will discuss these topics.