Algorithmic Resolution of Singularities and Nash multiplicity sequences

Resumen

In the study of algorithmic or constructive Resolution of Singularities, we make use of invariants that allow us to distinguish among different singular points of an algebraic variety. Attending to them, we choose the centers of a sequence of blow ups that will eventually lead to a resolution of the singularities of the initial variety (over characteristic zero fields). On the other hand, arc spaces are useful in the study of singularities, since they detect properties of algebraic varieties, including smoothness. They also let us define numerous invariants. In particular, the Nash multiplicity sequence is a non-increasing sequence of positive integers attached to an arc in the variety which stratifies the arc space. In this thesis, certain invariants of arcs are defined by means of this sequence. They also give rise to a series of invariants of singularities which turn out to be strongly related to those that we use for constructive resolution of singularities, for varieties defined over fields of characteristic zero. In fact, we show that one important version of Hironaka’s order (the most important invariant in constructive resolution) can be read in the arc space of the variety.