On asymptotic critical values and the Rabier Theorem

• Autorzy: Zbigniew Jelonek
• Języki publikacji: EN

Abstrakty

EN

Let X ⊂ kⁿ be a smooth affine variety of dimension n-r and let $f = (f₁,..., f_m): X → k^m$ be a polynomial dominant mapping. It is well-known that the mapping f is a locally trivial fibration outside a small closed set B(f). It can be proved (using a general Fibration Theorem of Rabier) that the set B(f) is contained in the set K(f) of generalized critical values of f. In this note we study the Rabier function. We give a few equivalent expressions for this function, in particular we compare this function with the Kuo function and with the (generalized) Gaffney function. As a consequence we give a direct short proof of the fact that f is a locally trivial fibration outside the set K(f) (i.e., that B(f) ⊂ K(f)). This generalizes the previous results of the author for $X = k^r$ (see [2]).

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Banach Center Publications
• Rocznik: 2004
• Tom: 65
• Numer: 1
• Strony: 125-133

Daty

wydano

2004

Twórcy

autor

Zbigniew Jelonek

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

Identyfikatory

DOI

10.4064/bc65-0-9