S. Encinas, A. Nobile and O. Villamayor

Given an algorithm for resolution of singularities that satisfies certain conditions ('a good algorithm'), natural notions of simultaneous algorithmic resolution, and of equi-resolution, for families of embedded schemes (parametrized by a reduced scheme $T$) are defined. It is proved that these notions are equivalent. Something similar is done for families of sheaves of ideals, where the goal is algorithmic simultaneous principalization. A consequence is that given a family of embedded schemes over a reduced $T$, this parameter scheme can be naturally expressed as a disjoint union of locally closed sets $T_j$, such that the induced family on each part $T_j$ is equi-resolvable. In particular, this can be applied to the Hilbert scheme of a smooth projective variety; in fact, our result shows that, in characteristic zero, the underlying topological space of any Hilbert scheme parametrizing embedded schemes can be naturally stratified in equi-resolvable families.

2000 Mathematical Subject Classification: 14E15, 14D99.

Keywords: resolution of singularities, algorithmic resolution, simultaneous resolution, Hilbert schemes.

E-mail: sencinas@maf.uva.es, nobile@marais.math.lsu.edu, villamayor@uam.es