Birational Finite Extensions of Mappings from a Smooth Variety

Marek Karaś¹

¹ Instytut Matematyki Uniwersytet Jagielloński Łojasiewicza 6 30-348 Kraków, Poland

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 57 (2009) no. 2, pp. 117-120

Résumé

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.

Détail

DOI : 10.4064/ba57-2-4

Keywords: present example finite mappings algebraic varieties rightarrow where subset subset rightarrow mathop gdeg mathop gdeg mathop gdeg means number points generic fiber sense result note improves result pure appl algebra where shown phenomenon occur subset subset geq subset similar example does exist