A Simplified Proof of Desingularization and Applications
A Simplified Proof of Desingularization and Applications
复制标题
去奇异化的简化证明和应用
DOI：
10.4171/rmi/425
复制
发表时间：
2002
影响因子：
1.2
通讯作者：
Orlando Villamayor
中科院分区：
文献类型：
--
作者：
Ana Maŕıa Bravo;S. Encinas;Orlando Villamayor
本文给出了特征零域上去奇化的一个简短而简化的证明，以及在代数几何中的其他问题上的各种应用(其中，研究了嵌入模式族的去奇化行为，以及一个比平中的去奇化更强的去奇化的公式)。我们的证明避免了使用Hilbert-Samuel函数和Hironaka的正规平坦性概念：首先，我们定义了理想的主化过程(即使理想可逆的过程)，然后我们证明了闭子方案X的去奇异化是通过使用与嵌入方案X相关的理想I(X)的主化过程来实现的。本文旨在介绍这个主题，重点介绍在这种新方法中使用的想法的动机，特别是应用，其中一些不是来自Hironaka的证明。
This paper contains a short and simplified proof of desingularization over fields of characteristic zero, together with various applications to other problems in algebraic geometry (among others, the study of the behavior of desingularization of families of embedded schemes, and a formulation of desingularization which is stronger than Hironaka's). Our proof avoids the use of the Hilbert-Samuel function and Hironaka's notion of normal flatness: First we define a procedure for principalization of ideals (i.e. a procedure to make an ideal invertible), and then we show that desingularization of a closed subscheme X is achieved by using the procedure of principalization for the ideal I(X) associated to the embedded scheme X. The paper intends to be an introduction to the subject, focused on the motivation of ideas used in this new approach, and particularly on applications, some of which do not follow from Hironaka's proof.