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# Wydawnictwa / Czasopisma IMPAN / Annales Polonici Mathematici / Wszystkie zeszyty

## Local characterization of algebraic manifolds and characterization of components of the set $S_f$

### Tom 75 / 2000

Annales Polonici Mathematici 75 (2000), 7-13 DOI: 10.4064/ap-75-1-7-13

#### Streszczenie

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

#### Autorzy

• Zbigniew Jelonek

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