# Local characterization of algebraic manifolds and characterization of components of the set ${S}_{f}$

Annales Polonici Mathematici (2000)

- Volume: 75, Issue: 1, page 7-13
- ISSN: 0066-2216

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topJelonek, Zbigniew. "Local characterization of algebraic manifolds and characterization of components of the set $S_f$." Annales Polonici Mathematici 75.1 (2000): 7-13. <http://eudml.org/doc/208387>.

@article{Jelonek2000,

abstract = {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 $f:ℂ^n→ℂ^m$.},

author = {Jelonek, Zbigniew},

journal = {Annales Polonici Mathematici},

keywords = {ℂ-uniruled variety; polynomial mappings; affine space; polynomial mapping},

language = {eng},

number = {1},

pages = {7-13},

title = {Local characterization of algebraic manifolds and characterization of components of the set $S_f$},

url = {http://eudml.org/doc/208387},

volume = {75},

year = {2000},

}

TY - JOUR

AU - Jelonek, Zbigniew

TI - Local characterization of algebraic manifolds and characterization of components of the set $S_f$

JO - Annales Polonici Mathematici

PY - 2000

VL - 75

IS - 1

SP - 7

EP - 13

AB - 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 $f:ℂ^n→ℂ^m$.

LA - eng

KW - ℂ-uniruled variety; polynomial mappings; affine space; polynomial mapping

UR - http://eudml.org/doc/208387

ER -

## References

top- [1] R. Hartshorne, Algebraic Geometry, Springer, New York, 1987.
- [2] Z. Jelonek, The set of points at which a polynomial map is not proper, Ann. Polon. Math. 58 (1993), 259-266. Zbl0806.14009
- [3] Z. Jelonek, Testing sets for properness of polynomial mappings, Math. Ann. 315 (1999), 1-35. Zbl0946.14039
- [4] K. Nowak, Injective endomorphisms of algebraic varieties, ibid. 299 (1994), 769-778. Zbl0803.14007

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