Rees algebras and resolution of singularities

Santiago Encinas; Orlando Villamayor

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arXiv · math/0702836

Rees algebras and resolution of singularities

Santiago Encinas·Orlando Villamayor

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Abstract

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 Włodarczyk. 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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Santiago Encinas, Orlando Villamayor. 2007-02-27. Rees algebras and resolution of singularities. https://arxiv.org/abs/math/0702836

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