On asymptotic critical values and the Rabier Theorem

Tom 65 / 2004

Zbigniew Jelonek Banach Center Publications 65 (2004), 125-133 MSC: Primary 51N10; Secondary 15A04. DOI: 10.4064/bc65-0-9


Let $X\subset k^n$ be a smooth affine variety of dimension $n-r$ and let $f=(f_1,\ldots, f_m): X \to k^m$ 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 $B(f)\subset K(f)$). This generalizes the previous results of the author for $X=k^r$ (see \cite{j}).


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

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