Discrete dynamical systems associated with the configuration space of 8 points in P^3(C)

A 3 dimensional analogue of Sakai’s theory concerning the relation between rational surfaces and discrete Painlev'e equations is studied. For a family of rational varieties obtained by blow-ups at 8 points in general position in ${\mathbb{P}}^3$, we define its symmetry group using the inner product that is associated with the intersection numbers and show that the group is isomorphic to the Weyl group of type $E_7^{(1)}$. By normalizing the configuration space by means of elliptic curves, the action of the Weyl group and the dynamical system associated with a translation are explicitly described. As a result, it is found that the action of the Weyl group on ${\mathbb{P}}^3$ preserves a one parameter family of quadratic surfaces and that it can therefore be reduced to the action on ${\mathbb{P}}^1\times {\mathbb{P}}^1$.