A Strengthening of Resolution of Singularities in Characteristic Zero

Let X be a closed subscheme embedded in a scheme W , smooth over a field k of characteristic zero, and let 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, π : W r → W, obtained as a composition of monoidal transformations, so that if X r ⊂ W r denotes the strict transform of X ⊂ W then: (1) the morphism π : W r → W is an embedded desingularization of X (as in Hironaka's Theorem); (2) the total transform of I (X) inOW rfactors as a product of an invertible sheaf of ideals L supported on the exceptional locus, and the sheaf of ideals defining the strict transform of X (that is,I ( X )OW r= L ⋅ 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.