On Algorithmic Equiresolution and Stratification of Hilbert Schemes

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Given an algorithm of resolution of singularities satisfying certain conditions (``good algorithms''), natural notions of simultaneous algorithmic resolution, or equiresolution, for families of embedded schemes (parametrized by a reduced scheme T) are proposed. It is proved that these conditions are equivalent. Something similar is done for families of sheaves of ideals, here 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 Tⱼ, such that the induced family on each part Tⱼ is equisolvable. 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 equiresolvable families.