Isolated plane curve singularities: a meeting point between algebra and topology.

Young researchers Math seminar

Speaker: Patricio Almirón

Abstract: The local study of isolated plane curve singularities is a vast research topic which has been developed during centuries. The main part of the talk will be to briefly overview some basic tools to work with topological and algebraic aspects about plane curve singularities. We will focus in the irreducible case, where life is easier; if time permits, we will try to mention something about the non-irreducible case, but the audience will probably be referred to a second part of this talk.

In the local study of an isolated plane curve singularity \(C\), an important algebraic object is the set of values \(S\) of interesection multiplicites of \(C\) with other curves. This set of values has a natural structure of semigroup, which in the irreducible case is a numerical semigroup (a semigroup over the natural numbers with finite complement). On the other hand, one can associate an algebraic link to the singularity which allows to classify their topological properties. The leitmotiv of the talk will be be to explain how the combinatorial properties of the semigroup of values can be used to understand the local topology of the associated link and viceversa.