  The subject is partial resolution of singularities. Given an algebraic
variety X (not necessarily equidimensional) in characteristic zero (or, more
generally, a pair (X,D), where D is a divisor on X), we construct a functorial
desingularization of all but stable simple normal crossings (stable-snc)
singularities, by smooth blowings-up that preserve such singularities. A
variety has stable simple normal crossings at a point if, locally, its
irreducible components are smooth and tranverse in some smooth embedding
variety. We also show that our main assertion is false for more general simple
normal crossings singularities.
