Rees algebras and resolution of singularities

Details Rees algebras and resolution of singularities Rees algebras and resolution of singularities

2007-02-27

arxiv.org/abs/math/0702836v1

Actas del XVI Coloquio Latinoamericano de Algebra, Biblioteca de la Revista Matematica Iberoamericana, Revista Matematica Iber

Mathematics

Algebraic Geometry

19 pages

Scientific paper

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.

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