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We show that every n-dimensional smooth algebraic variety X can be covered by Zariski open subsets which are isomorphic to closed smooth hypersurfaces in .
As an application we show that forevery (pure) n-1-dimensional ℂ-uniruled variety there is a generically-finite (even quasi-finite) polynomial mapping such that .
This gives (together with [3]) a full characterization of irreducible components of the set for generically-finite polynomial mappings .
We give a short proof of the Jacobian criterion of formal smoothness using the Lichtenbaum-Schlessinger cotangent complex.
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