标题： 一个强消奇定理 显示英文标题

标题： A strong desingularization theorem

摘要： 设 $X$ 是一个闭子概形，嵌入到一个在域 ${\bf k}$ 上光滑的概形 $W$ 中，并设 ${\mathcal I}(X)$ 是定义 $X$ 的理想层。假设 $X$ 的正则点集在 $X$ 中是稠密的。 我们证明存在一个适当的、双有理的同态$\pi: W_r\longrightarrow W$，它作为单参数变换的复合得到，使得如果$X_r\subset W_r$表示$X\subset W$的严格变换，则： 1) 同态$\pi:W_r\longrightarrow W$是$X$的嵌入消歧（如Hironaka定理中所述）；2){\em 总变换}在${\mathcal O}_{W_r}$中的${\mathcal I}(X)$因式分解为一个在例外支集上支持的可逆层理想${\mathcal L}$和定义$X$的严格变换的层理想乘积（即 ${\mathcal I}(X){\mathcal O}_{W_r}={\mathcal L}\cdot{\mathcal I}(X_r)$). 这个结果比Hironaka定理更强，事实上(2)是新的，并且在遵循Hironaka证明路线的消解中不成立，除非$X$是一个超曲面。 我们将说$W_r\longrightarrow W$定义了一个{\em 强 $X$的去奇异化}。 显示英文摘要

摘要： Let $X$ be a closed subscheme embedded in a scheme $W$ smooth over a field ${\bf k}$ of characteristic zero, and let ${\mathcal I}(X)$ be the sheaf of ideals defining $X$. Assume that the set of regular points of $X$ is dense in $X$. We prove that there exists a proper, birational morphism, $\pi: W_r\longrightarrow W$, obtained as a composition of monoidal transformations, so that if $X_r\subset W_r$ denotes the strict transform of $X\subset W$ then: 1) The morphism $\pi:W_r\longrightarrow W$ is an embedded desingularization of $X$ (as in Hironaka's Theorem); 2) The {\em total transform} of ${\mathcal I}(X)$ in ${\mathcal O}_{W_r}$ factors as a product of an invertible sheaf of ideals ${\mathcal L}$ supported on the exceptional locus, and the sheaf of ideals defining the strict transform of $X$ (i.e. ${\mathcal I}(X){\mathcal O}_{W_r}={\mathcal L}\cdot{\mathcal I}(X_r)$). This result is stronger than Hironaka's Theorem, in fact (2) is novel and does not hold for desingularizations which follow Hironaka's line of proof unless $X$ is a hypersurface. We will say that $W_r\longrightarrow W$ defines a {\em Strong Desingularization of $X$}.