Estudo local de curvas singulares via valorizações e semigrupos

Abstract

The main of this work is the local study of singular plane curves using valuations and semigroups of values. We have seen that the objects that correspond to the points of the curve are the valuations or, equivalently, the discrete valution rings. More precisely, let k be an algebraically closed field, C an irreducible non-singular projective plane curve and k(C) the rational function field of C. Then, there exists a bijection between the points of the curve C and the set of discrete valuations of the extension k(C)/k. We have also seen that in the case of singular curves this correspondence is not usually a bijection. We have studied semigroups of values associated with the local ring of some plane curves and we have also used the semigroup notions and relative ideals to characterize the torsion free modules of rank 1 on two examples of singular curves.