Invariants of singularities: from Nash to Hironaka

When we study 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.

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. As we will see, this sequence gives 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. Moreover, these invariants defined by means of arc spaces do not rely on the peculiarities of the characteristic zero case, so they pose interesting questions for the case of varieties defined over perfect fields, regardless of their characteristic.