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Weak normality and Lipschitz saturation for ordinary singularities. (English) Zbl 0591.14009
Let X be an algebraic variety defined over an algebraically closed field K. There are two homeomorphic varieties associated to X, the weak normalization \(X^*\) of X and the Lipschitz saturation \(\tilde X\) of X such that if \(X^{\#}\) is the normalization of X one has the following decomposition of the normalization morphism \[ \pi : X^{\#} \to X : X^{\#} \to X^* \to \tilde X \to X. \] If X is weakly normal (i.e \(X^*=X)\) then it is Lipschitz saturated (i.e. \(\tilde X=X)\). The converse of this assertion is false. The main result of this paper is the following: if X is obtained from a nonsingular projective variety by means of a linear projection from a center in general position, then \(X^*=\tilde X=X\). The main tool to prove this is to compare the weak normalization and the Lipschitz saturation using the double point scheme of the projection morphism.
Reviewer: M.Becheanu

MSC:
14E99 Birational geometry
14B05 Singularities in algebraic geometry
14E15 Global theory and resolution of singularities (algebro-geometric aspects)
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