A strong desingularization theorem

Abstract

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 $X$}.