  Some Weyl group acts on a family of rational varieties obtained by successive
blow-ups at $m$ ($m\geq n+2$) points in the projective space $\mpp^n(\mc)$. In
this paper we study the case where all the points of blow-ups lie on a certain
elliptic curve in $\mpp^n$. Investigating the action of Weyl group on the
Picard groups on the elliptic curve and on rational varieties, we show that the
action on the parameters can be written as a group of linear transformations on
the $(m+1)$-st power of a torus.
