Equivalence and resolution of singularities

This article provides a simple presentation of an algorithm to resolve singularities of algebraic varieties over fields of characteristic zero by means of a sequence of blowing ups with smooth centers contained in the set of points of maximum multiplicity. The algorithm uses primarily multiplicity, rather than the Hilbert–Samuel function, to control the resolution process, and it does not involve a local embedding into a smooth variety. The paper introduces a generalization of the usual notion of equivalence in the theory of resolution of singularities, which is important to justify an essential step in the construction of the algorithm.