Functoriality in resolution of singularities

Details Functoriality in resolution of singularities Functoriality in resolution of singularities

2007-02-13

arxiv.org/abs/math/0702375v1

Mathematics

Algebraic Geometry

27 pages

Scientific paper

Algorithms for resolution of singularities in characteristic zero are based on Hironaka's idea of reducing the problem to a simpler question of desingularization of an "idealistic exponent" (or "marked ideal"). How can we determine whether two marked ideals are equisingular in the sense that they can be resolved by the same blowing-up sequences? We show there is a desingularization functor defined on the category of equivalence classes of marked ideals and smooth morphisms, where marked ideals are "equivalent" if they have the same sequences of "test transformations". Functoriality in this sense realizes Hironaka's idealistic exponent philosophy. We use it to show that the recent algorithms for desingularization of marked ideals of Wlodarczyk and of Kollar coincide with our own, and we discuss open problems. This article is dedicated to Heisuke Hironaka for his 77th birthday, in celebration of "Kiju" -- joy and long life!

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