Rees algebras and resolution of singularities

TL;DR

This paper demonstrates that in characteristic zero, algorithms for desingularization produce the same Log-resolution for ideals with identical integral closures, extending the concept to Rees algebras and emphasizing the role of differential operators.

Contribution

It extends Log-resolution concepts from ideals to Rees algebras and proves that desingularization algorithms are consistent for Rees algebras with the same integral closure.

Findings

Algorithms of desingularization define the same Log-resolution for ideals with the same integral closure.

Rees algebras with the same integral closure undergo the same constructive resolution.

The interplay of integral closure with differential operators is key to the results.

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 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.