Weierstrass multiple points on algebraic curves and ramified coverings

Here we prove the following result on Weierstrass multiple points. Theorem:Fix integers k, g with k≥5 and g>4k. Then there exist a genus g, Riemann surface X and k points P ₁, …,P _k of X such that for all integers b ₁≥…≥b _k ≥0we have: . By Riemann-Roch the value given is the lowest one compatible withk, g and the inequalityh ⁰(X,O _X (P ₁+…+P _k ))≥2. Hence this theorem means that (P ₁, …,P _k ) is ak-ple Weierstrass set with the lowest weight possible compatible with the integersk andg. Using similar tools we prove a theorem on the non-gap sequence of a Weierstrass point onm-gonal curves and study theg _d ^r ’s on a generalk-sheeted covering of an irrational curve. Then we introduce and study a class of vector bundles on coverings of elliptic curves.