A theory of simultaneous resolution of singularities for families of embedded varieties (over a field of characteristic zero) parametrized by the spectrum of a suitable artinian ring, and compatible with a given algorithm of resolution, is presented. As usually, this a simple consequence of a similar theory for analogous families of basic objects, to which the main portion of this article is devoted.
"Algorithmic equiresolution of deformations of embedded varieties." Rev. Mat. Iberoamericana 25 (3) 995 - 1054, October, 2009.