Functoriality in Resolution of Singularities

By Edward Bierstone, Pierre D. Milman. 2007

88 citations recorded by the source corpus.

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 t

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/51a1a2851febef89df6a6059f8b42063fb915030

How researchers see this paper

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In Algebra and Number Theory

1. Yuri Prokhorov
2. Constantin Shramov
3. M. Temkin
4. Kiran S. Kedlaya
5. Ruochuan Liu

In Computational Theory and Mathematics

1. Yuri Prokhorov
2. Constantin Shramov
3. M. Temkin
4. Kiran S. Kedlaya

In Geometry and Topology

1. Yuri Prokhorov
2. Constantin Shramov
3. M. Temkin
4. Kiran S. Kedlaya
5. Ruochuan Liu
6. C. Birkar
7. V. Popov
8. Mattias Jonsson

Citations in the corpus

• Jordan property for groups of birational selfmaps (2013, background): “be the normalization of V˜ in the function field k(X). Then X˜ is a projective variety G-birational to X, and X˜ admits a biregular action of G. Taking a G-equivariant resolution of singularities (see [BM08]), one can assume that X˜ is smooth. Let Xbe a normal projective variety. By Cl(X) we denote t”
• Inseparable local uniformization (2008, background): “ver Q, see [Hir]. Moreover, it is achieved by blowing up regular centers, so one obtains fof a very special form. It was later proved that for varieties one can build ffunctorially, see, for example, [BM]. The case of general quasi-excellent schemes over Q was deduced in [Tem4] and [Tem6]. In positi”
• Functorial desingularization of quasi-excellent schemes in characteristic zero: the nonembedded case (2009, methodology): “stence of an (analog of) Taylor homomorphism A → A[[T]] due to Bierstone-Milman, this conjecture agrees with Bierstone-Milman philosophy. It is an interesting problem to generalize the algorithm from [BM2] accordingly to the conjecture. (iv) The above conjecture implies functorial desingularization ”