Nodal curves and Riccati solutions of Painlevé equations

Abstract

In this paper, we study Riccati solutions of Painlevé equations from a view point of geometry of Okamoto-Painlevé pairs (S, Y ). After establishing the correspondence between (rational) nodal curves on S − Y and Riccati solutions, we give the complete classification of the configurations of nodal curves on S − Y for each Okamoto-Painlevé pair (S, Y ). As an application of the classification, we prove the non-existence of Riccati solutions of Painlevé equations of types PI , PD 8 III and PD 7 III . We will also give a partial answer to the conjecture in [STT] and [T] that the dimension of the local cohomology H 1 Y red (S, ΘS(− log Y red )) is one.1991 Mathematics Subject Classification. 14D15, 34M55, 32G10.

Moduli spaces for linear differential equations and the Painlevé equations

“…The latter are related to Riccati solutions of the corresponding Painlevé equation. Since the Riccati curves on the Okamoto-Painlevé pairs are known ( [26]), one can now link each of the ten monodromy spaces R to an Okamoto-Painlevé pair and an extended Dynkin diagram (see Table 2.1). We remark, as done in [21], that for the caseD 6 there are two types of isomonodromic families corresponding to PV deg and PIII (D 6 ).…”

Dynamics on Character Varieties and Malgrange irreducibility of Painlevé VI equation

“…On H, the projection M(θ) → P 1 \ {0, 1, ∞} restricts to a regular rational fibration and the Painlevé equation restricts to a Riccati equation of hypergeometric type: we get a one parameter family of Riccati solutions. See [37,35,19] for a classification of singular points of S (A,B,C,D) and their link with Riccati solutions; they occur precisely when either one of the θ-parameter is an integer, or when the sum θ i is an integer. Since S (A,B,C,D) is affine, there are obviously no other complete curve in M t0 (θ) (see section 5.1).…”

Families of Okamoto–Painlevé pairs and Painlevé equations

“…For detail, we refer the reader to [8] and [ [11], Example 2.2]. Our purpose is to derive the regular algebraic vector fieldṽ |S−D on S − D, so from now on, we recall the explicit affine open covering not of π :…”

“…These works due to Okamoto, Takano et al and Sakai led to the series of works on the deformation theory of Okamoto-Painlevé pairs [11,12,[14][15][16], and this article. The Okamoto-Painlevé pair introduced in these papers is the pair of a generalized Halphen surface and its anti-canonical divisor.…”

“…For the coordinate system of the families of Okamoto-Painlevé pairs discussed in Sect. 3, we use Saito and Takebe's one [11] (essentially the same as Okamoto's one [8]) for typeẼ 8 , K. Takano et al's one [13] and [5] for typesẼ 7 , D 6 ,Ẽ 6 ,D 5 , andD 4 , and Sakai's one [10] for typesD 8 andD 7 .…”