A Strengthening of Resolution of Singularities in Characteristic Zero

Let X be a closed subscheme embedded in a scheme W, smooth overa field k of characteristic zero, and let I (X) be the sheafof ideals defining X. Assume that the set of regular pointsof X is dense in X. We prove that there exists a proper, birationalmorphism, : 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 : W_r W is an embedded desingularization ofX (as in Hironaka's Theorem); (2) the total transform of I (X) in factors as a product of an invertible sheaf of ideals L supportedon the exceptional locus, and the sheaf of ideals defining thestrict transform of X (that is, . Thus (2) asserts that we can obtain, in a simple manner, theequations defining the desingularization of X. 2000 MathematicalSubject Classification: 14E15.