Space of initial conditions for the four-dimensional Fuji-Suzuki-Tsuda system

A geometric study for an integrable 4-dimensional dynamical system so called the Fuji-Suzuki-Tsuda system is given. By the resolution of indeterminacy, the group of its B\"aklund transformations is lifted to a group of pseudo-isomorphisms between rational varieties obtained from $(P^1)^4$ by blowing-up along eight 2-dimensional subvarieties and four 1-dimensional subvarieties. The root basis is realised in the N\'eron-Severi bilattices. A discrete Painlev\'e system with quadratic degree growth is also realised as its translational element.