Discrete Dynamical Systems Associated with the Configuration Space of 8 Points in P 3 (C)

Abstract

A 3 dimensional analogue of Sakai's theory concerning the relation between rational surfaces and discrete Painlevé equations is studied. For a family of rational varieties obtained by blow-ups at 8 points in general position in 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 (1) 7 . 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 P 3 preserves a one parameter family of quadratic surfaces and that it can therefore be reduced to the action on P 1 × P 1 .

Space of initial conditions and geometry of two 4-dimensional discrete Painlevé equations

“…In this section we consider its non-autonomous version. That is, we consider Hamiltonian system of the form dq 1 dt = ∂I ∂p 1 , dp 1 dt = − ∂I ∂q 1 , dq 2 dt = ∂I ∂p 2 , dp 2 dt = − ∂I ∂q 2 (28) which is regular on a family of surfaces X a \D, where D is the support set of the singular anti-canonical divisor i m i D i . To find non-autonomous Hamiltonian, we use a technique used by Takano and his collaborators in [26,15] and by Sasano-Yamada in [25].…”

“…In the higher dimensional case the center of blowups is not necessarily a point but could be a subvariety of codimension two at least. Although some studies on symmetries of varieties or dynamical systems have been reported in the higher dimensional case, most of them consider only the case where varieties are obtained by blowups at points from the projective space [6,28,3]. One of few exceptions is [29], where varieties obtained by blowups along codimension three subvarieties from the direct product of a projective line (P 1 ) N were studied.…”

On the lattice-geometry and birational group of the six-point multi-ratio equation

“…Equations (39) allow the subset of variables in the initial data set (38), to be replaced by lattice parameters, α 0 , . .…”

“…, α i+2 , on which the group acts not rationally, but by pure permutation. The resulting partially-integrated system is generated by the group permutation action (Definition 2.1 and ( 40)) from the basic set of equations (39). Written in the demihypercube variables (Lemma 7.1), this is recognisable as the lattice Schwarzian KdV system: Proposition 7.3.…”

“…Case k = 1 of Definition 3.2 gives generators for a representation of the group of Coble [17,18]. This group is a basic element in the geometric framework of the Painlevé equations [13,19,38], and is a canvas for several extensions [39,20,40]. The connection is made by a change of variables.…”