Resolution except for minimal singularities I

The philosophy of the article is that the desingularization invariant together with natural geometric information can be used to compute local normal forms of singularities. The idea is used in two related problems: (1) We give a proof of resolution of singularities of a variety or a divisor, except for simple normal crossings (i.e., which avoids blowing up simple normal crossings, and ends up with a variety or a divisor having only simple normal crossings singularities). (2) For more general normal crossings (in a local analytic or formal sense), such a result does not hold. We find the smallest class of singularities (in low dimension or low codimension) with which we necessarily end up if we avoid blowing up normal crossings singularities. Several of the questions studied were raised by Kollar.Comment: 41 pages. Minor revisions; results unchanged. Reference adde

Functoriality in resolution of singularities

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!Comment: 27 page