Semigrupos e o teorema de Gorenstein para singularidades de curvas algébricas planas - Lannes, Andréa Maria Silva (2013)
The main goal of this dissertation is to present the Gorenstein Theorem for plane curve singularities. We consider two cases: firstly the local case when the singularity has only one branch and after the semilocal case when the singularity has several branches. In the local case the local equation is given by an irreducible series of k[[X, Y ]] and in the semilocal case it is given by a finite product of irreducible series wich are not pairwise associated. A local equation given by such a power series f is called an algebroid plane curve. The following are objects associated to an algebroid plane curve: The local ring O = O(f), its integral closure O˜ of O in its full ring of fractions and the conductor ideal of O˜ in O. We may say that these data encode all the algebraic / geometric informations of the algebroid plane curve (f). Gorenstein Theorem, that was proved in [Go] by D. Gorenstein states that, in both cases (local or semi-local), the codimension (as k-vector spaces) of the conductor ideal in the ring O is equal to the codimension of the ring O in the ring O˜. This provides us with a certain symmetry which is reflected in the semigroup associated to the algebroid plane curve (f). Thus, we also study the symmetry of semigroups of the natural numbers and relate them to the symmetry of the ring O in the local case.