Publication Date:
2003-03-01
Description:
Let $X$ be a closed subscheme embedded in a scheme $W$, smooth over a field ${ f k}$ 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 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 $X$ (that is, ${mathcal I}(X){mathcal O}_{W_r} = {mathcal L} cdot {mathcal I}(X_r)$).Thus (2) asserts that we can obtain, in a simple manner, the equations defining the desingularization of $X$.2000 Mathematical Subject Classification: 14E15.
Print ISSN:
0024-6115
Electronic ISSN:
1460-244X
Topics:
Mathematics
