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

TL;DR

This paper provides a geometric analysis of the initial conditions space for the 4D Fuji-Suzuki-Tsuda integrable system, revealing its structure through blow-ups and group actions, and connecting it to discrete Painlevé systems.

Contribution

It introduces a geometric framework for the Fuji-Suzuki-Tsuda system using blow-ups and pseudo-isomorphisms, and relates it to discrete Painlevé equations.

Findings

Group of Bäcklund transformations lifted to pseudo-isomorphisms.

Initial conditions space constructed via blow-ups of $(P^1)^4$.

Discrete Painlevé system realized as a translational element.

Abstract

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.