Weierstrass points, inflection points and ramification points of curves

Abstract

Let C be an integral curve of the smooth projective surface S and P $\epsilon$ C. Let $\pi:X\rightarrow C$ be the normalization and $Q\epsilon X$ with $\pi\left(Q\right)=P$. We are interested in the case in which Q is a Weierstrass point of X. We compute the semigroup N(Q, X) of non-gaps of Q when S is a Hirzebruch surface $F_{e}P\epsilon C_{reg}$ and P is a total ramification point of the restriction to C of a ruling $F_{e}\rightarrow P^{1}$. We study also families of pairs (X, Q) such that the first two integers of N( Q, X) are k and d. To do that we study families of pairs (P,C) with C plane curve, deg(C) =d, C has multiplicity d - k at P, C is unibranch at P and a line through P has intersection multiplicity d with C at P.