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

摘要

A 3 dimensional analogue of Sakai's theory concerning the relation betweenrational surfaces and discrete Painlev\'e equations is studied. For a family ofrational 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 isassociated with the intersection numbers and show that the group is isomorphicto the Weyl group of type $E_7^{(1)}$. By normalizing the configuration spaceby means of elliptic curves, the action of the Weyl group and the dynamicalsystem associated with a translation are explicitly described. As a result, itis found that the action of the Weyl group on ${\mathbb P}^3$ preserves a oneparameter family of quadratic surfaces and that it can therefore be reduced tothe action on ${\mathbb P}^1\times {\mathbb P}^1$.