Birational Finite Extensions of Mappings from a Smooth Variety

Volume 57 / 2009

Marek Karaś Bulletin Polish Acad. Sci. Math. 57 (2009), 117-120 MSC: 14Rxx, 14R10. DOI: 10.4064/ba57-2-4


We present an example of finite mappings of algebraic varieties $f:V\rightarrow W,$ where $V\subset {\bf k}^{n},W\subset {\bf k}^{n+1},$ and $F:{\bf k}^{n}\rightarrow {\bf k}^{n+1}$ such that $F|_{V}=f$ and $\mathop{\rm gdeg}F=1< \mathop{\rm gdeg}f$ ($\mathop{\rm gdeg}h$ means the number of points in the generic fiber of $h$). Thus, in some sense, the result of this note improves our result in J. Pure Appl. Algebra 148 (2000) where it was shown that this phenomenon can occur when $V\subset {\bf k}% ^{n},W\subset {\bf k}^{m}$ with $m\geq n+2.$ In the case $V,W\subset {\bf k}^{n}$ a similar example does not exist.


  • Marek KaraśInstytut Matematyki
    Uniwersytet Jagielloński
    Łojasiewicza 6
    30-348 Kraków, Poland

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