Cohomological Properties of Multiple Coverings of Smooth Projective Curves

Abstract

Let X, respectively C, be a smooth projective curve of genus g, respectively q, and f : X → C a degree k finite morphism. Set E:= f * (O X )/O C . Hence E is a rank k - 1 vector bundle on C with deg(E) = k q - k + 1 - g. Here we study the cohomological properties of E and in particular the integers h 0 (C, E(tP)), P ∈ C and t ∈ N. We use these integers to define the notion of f-Weierstrass points.