Space of initial conditions for the four-dimensional Garnier system revisited

A geometric study is given for the 4-dimensional Garnier system. By the resolution of indeterminacy, the group of its Bäklund transformations is lifted to a group of pseudo-isomorphisms between rational varieties obtained from ${\mathbb{P}}^2\times {\mathbb{P}}^2$ by 10 or 21 blow-ups. The root basis is discussed in the Néron-Severi bilattices for the space with 10 blow-ups.