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

• Autorzy: Zbigniew Jelonek
• Języki publikacji: EN

Abstrakty

EN

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 $f:ℂ^n→ℂ^m$.

Słowa kluczowe

EN

ℂ-uniruled variety; polynomial mappings; affine space

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Annales Polonici Mathematici
• Rocznik: 2000
• Tom: 75
• Numer: 1
• Strony: 7-13

Daty

wydano

2000

otrzymano

1999-06-25

poprawiono

2000-02-05

poprawiono

2000-10-16

Twórcy

autor

Zbigniew Jelonek

• Institute of Mathematics, Polish Academy of Sciences, Św. Tomasza 30, 31-027 Kraków, Poland

Bibliografia

• [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.
• [3] Z. Jelonek, Testing sets for properness of polynomial mappings, Math. Ann. 315 (1999), 1-35.
• [4] K. Nowak, Injective endomorphisms of algebraic varieties, ibid. 299 (1994), 769-778.