Local characterization of algebraic manifolds and characterization of components of the set Sf
Jelonek, Zbigniew
Annales Polonici Mathematici, Tome 75 (2000), p. 7-13 / Harvested from The Polish Digital Mathematics Library
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We show that every n-dimensional smooth algebraic variety X can be covered by Zariski open subsets Ui which are isomorphic to closed smooth hypersurfaces in ℂn+1. As an application we show that forevery (pure) n-1-dimensional ℂ-uniruled variety X⊂ℂm there is a generically-finite (even quasi-finite) polynomial mapping f:ℂn→ℂm such that X⊂Sf. This gives (together with [3]) a full characterization of irreducible components of the set Sf for generically-finite polynomial mappings f:ℂn→ℂm.

Publié le : 2000-01-01
EUDML-ID : urn:eudml:doc:208387 
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Jelonek, Zbigniew. Local characterization of algebraic manifolds and characterization of components of the set $S_f$
            . Annales Polonici Mathematici, Tome 75 (2000) pp. 7-13. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-apmv75z1p7bwm/

[00000] [1] R. Hartshorne, Algebraic Geometry, Springer, New York, 1987.

[00001] [2] Z. Jelonek, The set of points at which a polynomial map is not proper, Ann. Polon. Math. 58 (1993), 259-266. | Zbl 0806.14009

[00002] [3] Z. Jelonek, Testing sets for properness of polynomial mappings, Math. Ann. 315 (1999), 1-35. | Zbl 0946.14039

[00003] [4] K. Nowak, Injective endomorphisms of algebraic varieties, ibid. 299 (1994), 769-778. | Zbl 0803.14007