Rees algebras and resolution of singularities
By Santiago Encinas, Orlando Villamayor. 2007
49 citations recorded by the source corpus.
Embedded principalization of ideals in smooth schemes, also known as Log-resolutions of ideals, play a central role in algebraic geometry. If two sheaves of ideals, say $I_1$ and $I_2$, over a smooth scheme $V$ have the same integral closure, it is well known that Log-resolution of one of them induces a Log-resolution of the other. On the other hand, in case $V$ is smooth over a field of characteristic zero, an algorithm of desingularization provides, for each sheaf of ideals, a unique Log-resolution. In this paper we show that algorithms of desingularization define the same Log-resolution for two ideals having the same integral closure. We prove this result here by using the form of induction introduced by W{\l}odarczyk. We extend the notion of Log-resolution of ideals over a smooth scheme $V$, to that of Rees algebras over $V$; and then we show that two Rees algebras with the same integral closure undergo the same constructive resolution. The key point is the interplay of integral closure with differential operators.
Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/5dd66ab851d6976897d53177c89218d1a3189d42
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In Algebra and Number Theory
- Ana Bravo
- Herwig Hauser
- Orlando Villamayor
- Steven Dale Cutkosky
- J. Włodarczyk
- Brendan Hassett
- Donghoon Hyeon
In Computational Theory and Mathematics
- Ana Bravo
- Orlando Villamayor
- Herwig Hauser
- Steven Dale Cutkosky
- J. Włodarczyk
- Brendan Hassett
- Donghoon Hyeon
In Geometry and Topology
- Herwig Hauser
- Ana Bravo
- Orlando Villamayor
- Steven Dale Cutkosky
- J. Włodarczyk
- Santiago Encinas
- Donghoon Hyeon
- Brendan Hassett
Citations in the corpus
- On the problem of resolution of singularities in positive characteristic (Or: A proof we are still waiting for) (2009, methodology): “There is a novel approach to resolution by Villamayor and his collaborators Benito, Bravo and Encinas [Vi2, Vi3, BV, EV3 ].”
- Singularities in positive characteristic, stratification and simplification of the singular locus (2008, methodology): “We refer to Propositions 6.4, 6.5, 6.7 and Corollary 6.7 in [ 7 ].”