Local characterization of algebraic manifolds and characterization of components of the set ${\displaystyle S_f}$

We show that every n-dimensional smooth algebraic variety X can be covered by Zariski open subsets ${\displaystyle U_i}$ which are isomorphic to closed smooth hypersurfaces in ${\displaystyle {ℂ}^{n+1}}$. As an application we show that forevery (pure) n-1-dimensional ℂ-uniruled variety ${\displaystyle X\subset {ℂ}^m}$ there is a generically-finite (even quasi-finite) polynomial mapping ${\displaystyle f:{ℂ}^n\to {ℂ}^m}$ such that ${\displaystyle X\subset S_f}$. This gives (together with [3]) a full characterization of irreducible components of the set ${\displaystyle S_f}$ for generically-finite polynomial mappings ${\displaystyle f:{ℂ}^n\to {ℂ}^m}$.