J. Math. Kyoto Univ. (JMKYAZ) 00-0 (0000), 000-000

Nodal curves and Riccati solutions of Painlevé equations

By

Masa-Hiko Saito^{∗}and Hitomi Terajima^{∗∗}

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 P_{I}, P D̃_{8} ; III and P D̃_{7} ; III. We will also give a partial answer to the conjecture in [STT] and [T1] that the dimension of the local cohomology H_{Y_{red}}^{1}(S, Θ_{S}(− log Y_{red})) is one.

1. Introduction

A pair (S, Y) of a projective smooth surface S and an effective anticanonical divisor Y on S is called an Okamoto–Painlevé pair if it satisfies a suitable condition (see (2.1) in §2). In [STT], we established the theory of Okamoto–Painlevé pairs (S, Y) and characterize the Painlevé equations by means of the special deformation of Okamoto–Painlevé pairs. There exist 8 types of rational Okamoto–Painlevé pairs which correspond to the Painlevé equations. The types are classified by the types of the dual graphs of the configurations of Y, which are the affine Dynkin diagram of types R = D̃_{k}, 4 ≤ k ≤ 8, Ẽ_{l}, 6 ≤ l ≤ 8. For each R, we obtain the global family of Okamoto–Painlevé pairs

1991 Mathematics Subject Classification(s). 14D15, 34M55, 32G10 M.-H. Saito was partly supported by the Grant-in-Aid for Scientific Research (B-12440008)

and (Houga-138740023), JSPS. H. Terajima was partly supported by the Grant-in-Aid for JSPS Fellows, MEXT, Japan.

Communicated by N/A Additional infomation(s)

Revised Date Revised Dates

∗Kobe University

∗∗RIMS, Kyoto University

\begin{matrix}S_{R} & ↩ & D \\ π ↓ & ↙ & φ \\ M_{R} × B_{R} \end{matrix}  (1.1)

where B_{R} is an affine open subset of the t-affine line Spec C[t]. In [STT], the deformation with respect to the t-direction can be characterized by the local cohomology group H_{D}^{1}(S, Θ_{S}(− log D)) where D = Y_{red}. Furthermore we can show that the vector field \frac{∂}{∂t} has a unique lifting to a rational global vector field

ṽ ∈ H^{0}(S_{R}, Θ_{S_{R}}(− log D) ⊗ O_{S}(D))  (1.2)

which induces the Painlevé differential equations on S_{R} − D.

In the theory of Painlevé equations, it is important to determine all classical solutions, like algebraic solutions and Riccati solutions. (For the definition of classical solutions of Painlevé equations, see §1 in [U-W1]). In this direction, there are a considerable number of works by many authors. Here, we list up only a part of the references: (e.g., [DM], [Grm1], [Grm2], [Grm3], [Grm4], [Grm5], [Grm6], [Gr-Lu], [Gr-Ts], [Luk1], [Luk2], [Maz], [Mu1], [Mu2], [Ni], [Ohy], [O3], [U1], [U2], [U-W1], [U-W2], [V], [W1], [W2], [Y]). For example, in order to prove the irreducibility of the Painlevé equations, one has to determine the cases when the given Painlevé equations admit the Riccati solutions (cf. [U1], [U2], [NO], [U-W1], [U-W2]).

In this paper, a \underline{smooth} rational curve C ≃ P^{1} on a surface S with C^{2} = −2 is called a nodal curve or a (−2)-curve^{*1}. One of the main purpose of this paper is to characterize the Riccati solutions of Painlevé equations by means of geometry of nodal curves on S − Y_{red} for the corresponding rational Okamoto– Painlevé pairs (S, Y).

Since our characterization of Painlevé vector field ṽ (1.2) in [STT] is intrinsic, that is, coordinate free, so is the characterization of Riccati solutions.

Moreover we shall give the complete classification theorem (Theorem 3.1) of configurations of nodal curves on S − Y_{red} for all rational Okamoto–Painlevé pairs (S, Y) of non-fibered type and of additive type. As a corollary to Theorem (3.1), we can show that Painlevé equations P_{I}, P D̃_{7} ; III, P D̃_{8} ; III have no Riccati solutions for any parameters in the equations.

The following is a rough outline of this paper.

In §2, we characterize the Riccati solutions of the Painlevé equations by means of (−2)-curve (or nodal curve) C on S − Y_{red}. If for a given (α_{0}, t_{0}) ∈ M_{R} × B_{R}, the fiber S of π in (1.1) over (α_{0}, t_{0}) contains a nodal curve C ⊂ S − Y_{red}, we can extend the nodal curve C in the t-direction and obtain a family of nodal curves C ⟶ {α_{0}} × U where U is an (analytic or étale) open neighborhood of t_{0} in B_{R}. Then the restriction ṽ_{|C} is tangent to C which induces the Riccati equation on C. It seems that this approach is essentially equivalent to Umemura’s theory of invariant divisors for the Painlevé equations (cf. e.g., [U-W1]). However we believe that our approach gives a clearer geometric viewpoint of Riccati solutions of Painlevé equations.

*1We can always contract a nodal curve C ⊂ S to a singular point which is called the nodal surface singularity or the A_{1}-singularity.

In §3, we shall give the complete classification of configurations of nodal curves on S − Y for all rational Okamoto–Painlevé pairs (S, Y) of non-fibered type and of additive type. The classification is based on the structure theorem of the lattice induced by the intersection form on H^{2}(S, Z). We can show that the sub-lattice generated by the nodal curves C on S − Y is a sub-lattice of E_{8}^{−}, the unique even unimodular negative-definite lattice of rank 8. Then taking account into the sub-lattice generated by the irreducible components of Y, we can obtain the list of the possible configurations. For the existence of the possible configurations, we quote the Oguiso–Shioda’s classification theorem of singular fiber or Mordell–Weil group for rational elliptic surfaces. Note that a rational elliptic surface with a fixed fiber is a rational Okamoto–Painlevé pairs of fibered type in our terminology. Using the Oguiso–Shioda’s existence theorem and the deformation theory of Okamoto–Painlevé pairs, we shall show the existence of all possible configurations for some rational Okamoto–Painlevé pairs.

In §4, as a corollary to the classification theorem, we shall prove the nonexistence of Riccati solutions of the Painlevé equations of type R = P_{I}, P D̃_{7} ; III, P D̃_{8} ; III. Though there are other proofs for this result for P_{I} and P D̃_{7} ; III (e.g., [U1], [U2] and [Ohy]), our proof clarify the point that the obstruction to the existence of Riccati solutions lies in the topological conditions.

In §5, we give explicit examples of nodal curves and Riccati solutions of Painlevé equations associated to the nodal curves.

In §6, we shall give an example of the confluence of the Riccati solutions for R = Ẽ_{6}, (P_{IV}) and also the confluence of nodal curves. Moreover we give a remark on rational solutions coming from the intersection of two different Riccati solutions.

In Appendix A, as a corollary to Theorem 3.1, we shall give a partial answer to the Conjecture A.1 presented in [STT] and [T1] about the dimension of the local cohomology group.

2. (−2)-curves (Nodal curves) and Riccati solutions

In this section, we shall review the theory of Okamoto–Painlevé pairs and their relations to the Painlevé equations which were introduced in [STT].

2.1. Okamoto–Painlevé pairs

Definition 2.1. Let (S, Y) be a pair of a complex projective surface S and an effective anti-canonical divisor Y ∈ |−K_{S}| of S. Let Y = ∑_{i=1}^{r} m_{i}Y_{i} be the irreducible decomposition of Y. We call a pair (S, Y) an Okamoto–Painlevé pair if for all i, 1 ≤ i ≤ r,

Y · Y_{i} = deg[Y]_{|Y_{i}} = 0.  (2.1)

An Okamoto–Painlevé pair (S, Y) is called rational if S is a rational surface.

Remark 1. An Okamoto–Painlevé pair (S, Y) in Definition 2.1 is called a generalized Okamoto–Painlevé pair in [STT]. However, in this paper, we shall use this terminology. Note that in the original definition of an Okamoto– Painlevé pair (S, Y) in [STT] we assume that S − Y_{red} contains C^{2} as a Zariski open set and Y_{red} is a normal crossing divisor. (See also [Sa-Ta].)

2.2. Okamoto–Painlevé pairs and Painlevé equations

Let (S, Y) be a rational Okamoto–Painlevé pair with the irreducible decomposition Y = ∑_{i=1}^{r} m_{i}Y_{i} and set D = Y_{red} = ∑_{i=1}^{r} Y_{i}. Denote by M(Y) the sub-lattice of Pic(S) ≃ H^{2}(S, Z) generated by the irreducible components {Y_{i}}_{i=1}^{r}. With the bilinear form on M(Y) which is (−1) times the intersection pairing on S, M(Y) becomes a root lattice of affine type (cf. [Section 1, [STT]], [Sakai]). Let R(Y) denote the type of the root lattice. One can classify rational Okamoto–Painlevé pairs (S, Y) in terms of the type R(Y). (See [Section 1, [STT]], [Sakai]).

Not all types of rational Okamoto–Painlevé pairs correspond to the Painlevé equations. The Table 1 is the list of the types of Okamoto–Painlevé pairs which correspond to the Painlevé equations. We shall explain the meaning of the correspondence in Theorem 2.1. Note that classically, Painlevé equations were classified into 6 types, however now we should classify them into 8 types. Actually, the third Painlevé equations P_{III} can be classified further into 3 types P D̃_{6} ; III, P D̃_{7} ; III and P D̃_{8} ; III corresponding to the types of R = R(Y). The classical third Painlevé equations correspond to P D̃_{6} ; III, which form a two parameter family of equations. The equations P D̃_{7} ; III and P D̃_{8} ; III can be obtained by specializations of these parameters.

Okamoto–Painlevé pairs and Painlevé equations

R = R(Y)  Ẽ_{8}  Ẽ_{7}  D̃_{8}  D̃_{7}  D̃_{6}  Ẽ_{6}  D̃_{5}  D̃_{4}

Painlevé equation P_{I} P_{II} P D̃_{8} ; III P D̃_{7} ; III P D̃_{6} ; III P_{IV} P_{V} P_{V I}

Table 1.

Here we shall recall one more important definition (cf. [Section 1, [STT]]).

Definition 2.2. A rational Okamoto–Painlevé pair (S, Y) will be called of fibered type if there exists an elliptic fibration f : S ⟶ P^{1} such that f^{∗}(∞) = Y as divisors. We say that a rational Okamoto–Painlevé pair is of non-fibered type if (S, Y) is not of fibered type.

The following theorem (cf. [Proposition 5.1, Theorem 6.1, [STT]]) explains how one can give correspondences between rational Okamoto–Painlevé pairs and Painlevé equations in Table 1.

Theorem 2.1 (Proposition 5.1. [STT]). Let R = R(Y) be one of types of the root systems in Table 1 (i.e., R = D̃_{i}, 4 ≤ i ≤ 8 or Ẽ_{j}, 6 ≤ j ≤ 8) and let r be the number of irreducible components of D = Y_{red} and set s = s(R) = 9−r. Then there exist affine open subschemes M_{R} ⊂ C^{s} = Spec C[α_{1}, ⋯, α_{s}], B_{R} ⊂ C = Spec C[t], and the following commutative diagram satisfying the conditions below:

\begin{matrix}S & ↩ & D \\ π ↓ & ↙ & φ \\ M_{R} × B_{R} & & . \end{matrix}  (2.2)

1. S is a smooth quasi-projective manifold and D is a divisor with normal crossing of S. Moreover π is a smooth and projective morphism and φ is a flat morphism such that the above diagram is a deformation of non-singular pairs of projective surfaces and normal crossing divisors in the sense of Kawamata [Kaw].

2. There is a rational relative 2-form

ω_{S} ∈ Γ(S, Ω_{S/M_{R}×B_{R}}^{2}(∗D))  (2.3)

which has poles only along D. If we denote by Y the pole divisor of ω_{S}, then for each point (α, t) ∈ M_{R} × B_{R}, (S_{α,t}, Y_{α,t}) is a rational Okamoto–Painlevé pair of type R = R(Y) and Y_{red} = D.

3. There is a unique global rational vector field

ṽ ∈ Γ(S, Θ_{S}(− log D) ⊗ O_{S}(D))  (2.4)

on S which is a lift of \frac{∂}{∂t}, that is, π_{∗}(ṽ) = \frac{∂}{∂t}. Moreover the restriction of ṽ to S − D gives a regular algebraic vector field which corresponds to the Painlevé equation of type R. We call the systems of differential equations determined by the vector field ṽ the Painlevé system of type R. (See (2.10) below).

We can state more about the family in (2.2) as follows.

1. The family is semi-universal at each point (α, t) ∈ M_{R} × B_{R}, that is, the Kodaira–Spencer map

ρ : T_{α,t}(M_{R} × B_{R}) ⟶ H^{1}(S_{α,t}, Θ_{S_{α,t}}(− log D_{α,t}))  (2.5)

is an isomorphism. For a point (α, t) ∈ M_{R} × B_{R} at which the corresponding Okamoto–Painlevé pair is of non-fibered type, one can obtain the following commutative diagram:

0 ↑

0 → H_{D_{α,t}}^{1}(S_{α,t}, Θ_{S_{α,t}}(− log D_{α,t}))  ↪ H^{1}(S_{α,t}, Θ_{S_{α,t}}(− log D_{α,t}))
↑  ||
H^{0}(D_{α,t}, Θ_{S_{α,t}}(− log D) ⊗ N_{D_{α,t}}) ≃ C · ρ( \frac{∂}{∂t}) ↪ H^{1}(S_{α,t}, Θ_{S_{α,t}}(− log D_{α,t}))
≀ ↑  ≀ ↑ ρ
T_{α,t}(B_{R}) ≃ C · \frac{∂}{∂t} ↪ T_{α,t}(M_{R} × B_{R})
↑  ↑
0  0

(2.6)

2. Let M_{R} and B_{R} denote the affine coordinate rings of M_{R} and B_{R} respectively so that M_{R} = Spec M_{R} and B_{R} = Spec B_{R}. (Note that M_{R} and B_{R} are obtained by some localizations of C[α_{1}, ⋯, α_{s}] and C[t] respectively). There exists an affine open covering {Ũ_{i}}_{i=1}^{l+k} of S such that for each i

Ũ_{i} ≃ Spec ((M_{R} ⊗ B_{R})[x_{i}, y_{i}, \frac{1}{f_{i}(x_{i}, y_{i}, α, t)} ]) ⊂ Spec C[α, t, x_{i}, y_{i}] ≃ C^{s+3} ≃ C^{12−r}.

(2.7)

Here f_{i}(x_{i}, y_{i}, α, t) is a polynomial in (M_{R} ⊗ B_{R})[x_{i}, y_{i}]. Moreover, we may assume that S − D can be covered by {Ũ_{i}}_{i=1}^{l}, and for each i, the restriction of the rational 2-form ω_{S} can be written as

ω_{S|Ũ_{i}} = \frac{dx_{i} ∧ dy_{i}}{f_{i}(x_{i}, y_{i}, α, t)^{m_{i}}} .  (2.8)

3. By using the local coordinates of S − D, the global rational vector field ṽ on S obtained in (2.4) can be written on each open set Ũ_{i} for 1 ≤ i ≤ l (corresponding to the open covering of S − D) as

ṽ_{|Ũ_{i}} = \frac{∂}{∂t} − θ_{i} = \frac{∂}{∂t} − η_{i} \frac{∂}{∂x_{i}} − ζ_{i} \frac{∂}{∂y_{i}}  (2.9)

where θ_{i} = η_{i} \frac{∂}{∂x_{i}} + ζ_{i} \frac{∂}{∂y_{i}} is a regular algebraic vector field on Ũ_{i}. This explicit expression of ṽ_{|Ũ_{i}} gives a system of differential equations

\{ \begin{matrix}\frac{dx_{i}}{dt} & = & −η_{i}(x_{i}, y_{i}, α, t) \\ \frac{dy_{i}}{dt} & = & −ζ_{i}(x_{i}, y_{i}, α, t) \end{matrix}  (2.10)

which is equivalent to the Painlevé equation of type R.

Remark 2. One can show that the deformation corresponding to ρ(\frac{∂}{∂t}) preserves the relative rational 2-form ω_{S} in (2.3). This fact explains the reason why the systems of differential equations in (2.10) can be written in Hamiltonian systems. For more details, see [§6, [STT]].

2.3. Riccati equations

Let U ⊂ C be an open complex domain (in analytic topology) with a local analytic coordinate t and a(t), b(t), c(t) holomorphic functions defined in U.

Consider a Riccati equation

x′ = a(t)x^{2} + b(t)x + c(t).  (2.11)

By the change of unknown

x = −\frac{1}{a(t)} \frac{d}{dt} log(u) = −\frac{1}{a(t)} \frac{u′}{u} ,  (2.12)

the equation (2.11) is transformed into the linear equation

u′′ − [\frac{a′(t)}{a(t)} + b(t)]u′ + a(t)c(t)u = 0.  (2.13)

Therefore the movable singularities of the solution x(t) = −\frac{1}{a(t)} \frac{u′}{u} of (2.11) are only poles. This condition is called the Painlevé property for an algebraic ordinary differential equation. (Cf. [3.1, Ch. 3, [IKSY]]).

Remark 3. Riccati equations above are defined in the space P^{1} × U with the coordinates (x, t). The equation (2.11) is equivalent to a rational global vector field on P^{1} × U as

ṽ = \frac{∂}{∂t} + [a(t)x^{2} + b(t)x + c(t)]\frac{∂}{∂x}.  (2.14)

By the coordinate change u = \frac{1}{x}, ṽ can be transformed into the form

ṽ = \frac{∂}{∂t} − [a(t) + b(t)u + c(t)u^{2}]\frac{∂}{∂u}.

This shows that the vector field ṽ is holomorphic even at x = ∞, hence ṽ is a global holomorphic vector field on P^{1} × U. (Conversely, one can show that any holomorphic vector field on P^{1} × U which is a lift of \frac{∂}{∂t} can be written as in (2.14)). Therefore the space P^{1} × U can be considered as the space of initial conditions for the Riccati equation above.

2.4. Nodal curves on Okamoto–Painlevé pairs and Riccati equations

Let (S, Y) be a rational Okamoto–Painlevé pair of type R = R(Y) corresponding to Painlevé equations of type R. Then, as we see in Theorem 2.1, one can construct a global rational vector field ṽ on the semi-universal deformation family of (S, Y) which gives the Painlevé equation of type R.

In what follows, we will show that Painlevé equations can be reduced to the Riccati equations if and only if the corresponding rational Okamoto–Painlevé pair (S, Y) contains P^{1} on S − Y_{red}. Roughly speaking, we have the following correspondences.

Painlevé equations ⇔ Special deformations of Okamoto– Painlevé pairs (S, Y)

∪  ∪
Riccati equations ⇔ Nodal curves C ≃ P^{1} ⊂ S − Y_{red}

(2.15)

In order to explain this scheme more explicitly, let us consider the Hamiltonian systems of the Painlevé equation of type Ẽ_{6} (= P_{IV}) with two auxiliary parameters κ_{0}, κ_{∞}.

\{ \begin{matrix}\frac{dx_{0}}{dt} & = & 4x_{0}y_{0} − x_{0}^{2} − 2tx_{0} − 2κ_{0} \\ \frac{dy_{0}}{dt} & = & −2y_{0}^{2} + 2(x_{0} + t)y_{0} − κ_{∞} \end{matrix} .  (2.16)

When κ_{0} = 0, if we set x_{0} ≡ 0, the first equation of the system (2.16) is automatically satisfied, and the second equation can be reduced to the equation

\frac{dy_{0}}{dt} = −2y_{0}^{2} + 2ty_{0} − κ_{∞},  (2.17)

which is nothing but a Riccati equation. One can easily check that {x_{0} = 0} defines a smooth P^{1} on S − Y_{red}. (See §4).

Note that if C ⊂ S −Y_{red} is a smooth irreducible rational curve in S −Y_{red}, we see that K_{S} · C = −Y · C = 0, hence, by the adjunction formula, we have

C^{2} = K_{S} · C + C^{2} = −2.

Therefore a smooth irreducible rational curve C ⊂ S − Y_{red} is always a nodal curve or a (−2)-curve .

The following proposition gives a characterization of Riccati equations obtained from the Painlevé equations in terms of rational nodal curves on Okamoto–Painlevé pair (S, Y). (See Figure 1).

Proposition 2.1. Under the same notation as in Theorem 2.1, let us consider the family π : S ⟶ M_{R} × B_{R} of the Okamoto–Painlevé pairs of type R in ( 2.2).

1. Assume that for a point t′_{0} = (α_{0}, t_{0}) ∈ M_{R}×B_{R}, there exists a smooth rational curve C ⊂ S_{(α_{0},t_{0})} − D_{(α_{0},t_{0})}. Then there exists an (analytic or étale) open neighborhood U of t_{0} of B_{R} satisfying the following conditions.

(a) There exist a flat family of rational curves φ : C ⟶ {α_{0}} × U and an inclusion ι : C ↪ S − D|_{{α_{0}}×U} such that the following diagram is commutative:

\begin{matrix}C & ↪ & C & ↪^{ι} & & S − D|_{\{α_{0}\}×U} \\ ↓ & & φ ↓ & ↙ & π \\ (α_{0}, t_{0}) & ∈ & \{α_{0}\} × U \end{matrix}  (2.18)

(b) The restriction of the vector field ṽ ∈ Γ(S, Θ_{S}(− log D) ⊗ O_{S}(D)) in ( 2.4) to C is tangent to C, that is,

ṽ|_{C} ∈ H^{0}(C, Θ_{C}).  (2.19)

Moreover ṽ_{|C} defines a Riccati equation.

2. Conversely, assume that the restriction of Painlevé equation ṽ_{|S′} to the family π′ : S′ := S_{|{α_{0}}×B_{R}} ⟶ {α_{0}} × B_{R} can be reduced to a Riccati equation on an open neighborhood {α_{0}}×U of a point (α_{0}, t_{0}) ∈ {α_{0}}×B_{R}. Then there exist a family of rational nodal curves C ⟶ {α_{0}} × U on π′ : S′ − D′ ⟶ {α_{0}} × U.

Proof. Let us set B_{α_{0}} = {α_{0}}×B_{R} ↪ M_{R} ×B_{R}, t′_{0} = (α_{0}, t_{0}). Restricting the family S ⟶ M_{R} × B_{R} to B_{α_{0}}, we obtain a smooth projective family of surfaces:

π′ : S′ := S_{|B_{α_{0}}} ⟶ B_{α_{0}}.

Moreover, we set S_{t′_{0}} = π′^{−1}(t′_{0}). Fix a relatively ample line bundle H for π′ : S′ ⟶ B_{α_{0}}. Consider the connected component T of the Hilbert scheme Hilb(S′/B_{α_{0}}) which contains a point [C] and let C ⟶ T denote the corresponding universal family. (Since π′ is projective and smooth, the universal family τ : C ⟶ T exists (cf. [Theorem 1.4, Ch. I, [Kol]]).)

Moreover we have a natural morphism ϕ : T → B_{α_{0}} and a natural inclusion ι : C ↪ T ×_{B_{α_{0}}} S′, so that τ : C ⟶ T can be factorized into τ = p_{1} ∘ ι where p_{1} denotes the first projection.

Let (Q, m_{Q}) be the local ring of T at [C]. Then from [Theorem 2.10, Ch. I, [Kol]], one can see the following:

1. The O_{B_{α_{0}} ,t′_{0}} -algebra Q can be written as the quotient of a local O_{B_{α_{0}} ,t′_{0}} -algebra P, where

Spec P ⟶ B_{α_{0}}

is smooth of relative dimension d = dim H^{0}(C, N_{C/S_{t′_{0}}} ).

2. The kernel K = ker[P → Q] is generated by dim Obs(C) elements where Obs(C) denotes the space of obstructions.

Since C ⊂ S_{t′_{0}} is a (−2)-curve, we see that N_{C/S_{t′_{0}}} ≃ O_{C}(−2), and hence we have H^{0}(C, N_{C/S_{t′_{0}}} ) = H^{0}(P^{1}, O_{P^{1}}(−2)) = {0}. Therefore Spec P ⟶ B_{α_{0}} is smooth of relative dimension 0. Now we claim that:

Claim: Obs(C) = {0}.  (2.20)

Assuming the claim, we see that

P ≃ Q ≃ O_{B_{α_{0}} ,t′_{0}},

hence this implies that T is a smooth variety of dimension 1 at the point [C] and the morphism ϕ : T ⟶ B_{α_{0}} is also an isomorphism near [C] (étale or analytic) locally. Hence we obtain an open neighborhood U′ of [C] in T on which the morphism ϕ induces the isomorphism ϕ_{|U′} : U′ ⟶^{≃} ϕ(U′) ⊂ B_{α_{0}}. It is clear that U′ = α_{0} × U for some open neighborhood of t_{0} in B_{R} and the restriction of the family C ⟶ T to U′ gives a family of rational curves C ⟶ {α_{0}} × U which is a deformation of the rational curve C in S_{t′_{0}}.

Now we show the claim (2.20).

From [Proposition 2.14, Ch. 1., [Kol]], one see that the space of the obstructions Obs(C) lies in H^{1}(C, N_{C/S_{t′_{0}}} ). Consider the natural homomorphisms of cohomology groups

H^{1}(S_{t′_{0}}, Θ_{S_{t′_{0}}} ) ⟶^{ν} H^{1}(C, Θ_{S_{t′_{0}} |C}) ⟶^{μ} H^{1}(C, N_{C/S_{t′_{0}}} ).

Combining the Kodaira-Spencer homomorphism ρ : T_{B_{α_{0}} ,t′_{0}} ⟶ H^{1}(S_{t′_{0}}, Θ_{S_{t′_{0}}} ), it is easy to see that

Obs(C) = μ ∘ ν ∘ ρ(T_{B_{α_{0}} ,t′_{0}}).  (2.21)

For simplicity, we set S = S_{t′_{0}}, Y = Y_{t′_{0}}, D = Y_{red} = ∑_{i=1}^{r} Y_{i}.

Since C ⊂ S − D, we see that

Θ_{S}(− log(D + C))_{|D} ≃ Θ_{S}(− log D)_{|D}.

Therefore we have the following exact sequence

0 ⟶ Θ_{S}(− log(D+C)) ⟶ Θ_{S}(− log(D+C))(D) ⟶ Θ_{S}(− log D)⊗N_{D/S} ⟶ 0.

For an Okamoto–Painlevé pair (S, Y) of non-fibered type, we have (cf. [Proposition 2.1, [STT]]) H^{0}(S, Θ_{S}(− log(D + C))(D)) = {0}. Hence, this gives an injective homomorphism

0 → H^{0}(D, Θ_{S}(− log D) ⊗ N_{D/S}) ↪ H^{1}(S, Θ_{S}(− log(D + C)).  (2.22)

We also have the following commutative diagram of sheaves (cf. [Lemma 2.1, [STT]]):

0  0  0
↓  ↓  ↓
0 ⟶ Θ_{S}(− log(D + C)) ⟶ Θ_{S}(− log D) ⟶ N_{C/S} ⟶ 0
↓  ↓  ↓
0 ⟶  Θ_{S}(− log C)  ⟶  Θ_{S}  ⟶ N_{C/S} ⟶ 0
↓  ↓  ↓
0 ⟶  ⊕_{i=1}^{r}N_{Y_{i}/S}  ⟶ ⊕_{i=1}^{r}N_{Y_{i}/S} ⟶ 0 ⟶ 0
↓  ↓  ↓
0  0  0  .

Since N_{Y_{i}/S} = O_{Y_{i}}(−2) and N_{C/S} = O_{C}(−2), we have the inclusions

H^{1}(S, Θ_{S}(− log(D + C))) ↪ H^{1}(S, Θ_{S}(− log C)) ↪ H^{1}(S, Θ_{S}).

Combining this and (2.22), we see that

H^{0}(D, Θ_{S}(− log D) ⊗ N_{D/S}) ↪  H^{1}(S, Θ_{S}(− log(D + C)))
∩

ker[μ ∘ ν : H^{1}(S, Θ_{S}) ⟶ H^{1}(C, N_{C/S})]. (2.23)

From (2.6), we have

ρ(T_{B_{α_{0}} ,t′_{0}}) ≃ H^{0}(D, Θ_{S}(− log D) ⊗ N_{D/S}),

and hence

μ ∘ ν ∘ ρ(T_{B_{α_{0}} ,t′_{0}}) = {0}.

Together with (2.21), this shows the claim (2.20).

Next, let us consider the family

\begin{aligned}C & ↪ & S′_{|U} \\ & ↘ & ↓ π \\ & & U. \end{aligned}  (2.24)

Since D ∩ C = ∅, we have Θ_{S′|C} = Θ_{S′}(− log D) ⊗ O_{S′}(D)_{|C}, and hence we obtain the following exact sequence:

0 ⟶ Θ_{C} ⟶ Θ_{S′}(− log D) ⊗ O_{S′}(D)_{|C} ⟶ N_{C/S′} ⟶ 0.

Since N_{C/S′|S′_{t}} = N_{C_{t}/S′_{t}} = O_{C_{t}}(−2), we can show that π_{∗}(N_{C/S′}) = {0}. Then we have Γ(C, N_{C/S′}) = {0}. This implies that

H^{0}(C, Θ_{C}) ≃ H^{0}(S′, Θ_{S′}(− log D) ⊗ O_{S′}(D)_{|C}).

Hence ṽ_{|C} ∈ H^{0}(C, Θ_{C}).

Moreover, we may assume that C ⟶ U is a trivial P^{1}-bundle, that is, C ≃ P^{1} ×U analytically. Since ṽ_{|C} defines a holomorphic vector field on P^{1} ×U, it is easy to see that ṽ_{|C} is equivalent to a Riccati equation (cf. Remark 3).

The second assertion is now obvious, because the space of initial conditions of a Riccati equation must be a family of P^{1}. (Cf. Remark 3).

Remark 4 (Global deformation of a (−2)-curve C). Let us consider the connected component T of the Hilbert scheme Hilb(S′/B_{α_{0}}) which contains a point [C] and the corresponding universal family τ : C ⟶ T in the proof of Proposition 2.1. The argument in the proof shows that dim T = 1 and the natural morphism

ϕ : T ⟶ B_{α_{0}}

is projective, and hence surjective. We see that ϕ is a finite morphism of degree d ≥ 1. Assume that ϕ is an isomorphism, i.e., d = 1. Then we have the global family of rational curves C ⊂ S′ over the affine curve B_{α_{0}}:

\begin{aligned}C & ⊂ & C & ↪ & S′ \\ ↓ & & ↓ & & ↓ \\ t′_{0} & ∈ & T & = & B_{α_{0}} \end{aligned} .  (2.25)

12 Masa-Hiko Saito and Hitomi Terajima

SB_{R} t C ṽ_{|C}

Figure 1. Nodal curves and Riccati equations for Ẽ_{6} (P_{IV})

Then the vector field ṽ_{|C} becomes an algebraic regular vector field on C and defines a Riccati equation over the affine algebraic curve B_{α_{0}}. In this case, we call the differential equation defined by ṽ_{|C} the Riccati equation associated to the rational curve C(⊂ S − D).

We do not know whether the case with d > 1 occurs or does not occur. However, if ϕ : T ⟶ B_{α_{0}} is of degree d > 1, we see that ϕ^{−1}(ϕ([C])) consists of d rational curves of S_{t′_{0}} C_{1} := C, C_{2}, ⋯, C_{d} which are in the flat family of rational curves in S′ parametrized by a connected variety T.

In §2, we see that there exists an Okamoto–Painlevé pair (S, Y) which contains more than one rational curves C_{i} ⊂ S − D, i ≥ 2.

Definition 2.3. Under the same notation and assumptions in Proposition 2.1, we call the differential equations determined by the vector field ṽ|_{C} in (2.14) Riccati equation associated with the rational curve C ⊂ S − Y_{red}. Moreover we call a solution of the Riccati equation ṽ|_{C} a Riccati solution of the Painlevé system (associated with C ⊂ S − Y_{red}). (Note that all solutions of ṽ|_{C} remain in the family of rational curves in (2.18)).

3. Classification of (−2)-rational curves (nodal rational curves) on S − D

Let (S, Y) be a rational Okamoto–Painlevé pair of non-fibered type which corresponds to a Painlevé equation (cf. Table 1).

In this section, we will classify all configurations of (−2)-curves on S − D for a rational Okamoto–Painlevé pair (S, Y) of non-fibered type. The classification of the configurations are based on the similar classification for rational Okamoto–Painlevé pairs (S, Y) of fibered type with the elliptic fibration f : S ⟶ P^{1} and some deformation arguments.

3.1. Notations and the Result

Let S be a projective smooth surface over C. We denote by Div(S) the free abelian group generated by all irreducible curves on S. Let ∼_{a} and ∼ denote the algebraic equivalence and the linear equivalence of divisors respectively. We define the Néron–Severi group and the Picard group of S by

NS(S) = Div(S)/ ∼_{a},  (3.1)
Pic(S) = Div(S)/ ∼ .  (3.2)

In what follows, we assume that S is a rational surface. Then we have the natural isomorphisms

Pic(S) ≃ NS(S) ≃ H^{2}(S, Z),  (3.3)

and these groups are free Z-modules of rank b_{2}(S). For any divisor C, we also denote by the same letter C the class of the divisor in NS(S) ≃ H^{2}(S, Z). Moreover C = D means that the two divisors are linear equivalent to each other. We can consider the lattice structure on these free Z-modules by the intersection form < , > on NS(S) or equivalently by the cup product on H^{2}(S, Z). Let E_{8} be the unique even unimodular positive-definite lattice of rank 8. For a lattice L = (L, < , >), we denote by L^{−} = (L, (−1)× < , >), the opposite lattice of L. Note that the opposite lattice E_{8}^{−} of E_{8} is negative-definite.

Let (S, Y) be a rational Okamoto–Painlevé pair and let

Y = ∑_{i=1}^{r} m_{i}Y_{i}  (3.4)

be the irreducible decomposition of Y. Since S is a rational surface with b_{2}(S) = rank H^{2}(S, Z) = 10, by the Hodge index theorem, the bilinear form < , > on H^{2}(S, Z) can be written as the diagonal matrix (1, \underbrace{−1, ⋯, −1}_{9}). The sub-lattice

M(Y) generated by {Y_{i}}_{i=1}^{r} in H^{2}(S, Z) is a root lattice of an affine type, say R = R(Y). Since S is not relatively minimal, S contains a (−1)-rational curve O on S. Then by the adjunction formula, one has Y · O = −K_{S} · O = 1. Hence, there exists a i_{0}, 1 ≤ i_{0} ≤ r such that m_{i_{0}} = 1 and Y_{i_{0}} · O = 1. By renumbering i, we may assume that i_{0} = 1. Define the sub-lattice by

M′(Y) = ⟨Y_{2}, ⋯, Y_{r}⟩_{Z} ⊂ M(Y),  (3.5)

which is a root lattice of classical type R′. For example, if R = D̃_{4}, then R′ = D_{4}. Let M(S − Y_{red}) be the sub-lattice H^{2}(S, Z) generated by all (−2)- curves C on S − Y. Note that we have the orthogonal sum

M′(Y) ⊕ M(S − Y_{red}) ⊂ H^{2}(S, Z).  (3.6)

Lemma 3.1. Assume that (S, Y) is of non-fibered type. Then M′(Y) ⊕ M(S − Y_{red}) is a root sub-lattice of E_{8}^{−}.

Proof. The sub-lattice ⟨Y, O⟩_{Z} generated by Y and O has the intersection

matrix ( \begin{matrix}0 & 1 \\ 1 & −1 \end{matrix} ).

Then the orthogonal complement ⟨Y, O⟩^{⊥} in H^{2}(S, Z) is an even, negativedefinite unimodular lattice of rank 8, which is isomorphic to the root lattice E_{8}^{−}. (Since K_{S} = −Y, the adjunction formula implies that ⟨Y, O⟩^{⊥} is even). Since Y · O = 1, we see that the orthogonal complement ⟨Y ⟩^{⊥} is given by

⟨Y ⟩^{⊥} ≃ ⟨Y, O⟩^{⊥} ⊕ ZY ≃ Ẽ_{8}^{−}.

Since M(S − Y_{red}) is generated by (−2)-curves on S − Y_{red}, we see that M(S − Y_{red}) ⊂ ⟨Y ⟩^{⊥}. Moreover by definition of Okamoto–Painlevé pair (cf. (2.1)), M′(Y) ⊂ ⟨Y ⟩^{⊥}. (In fact, we have M′(Y) ⊂ ⟨Y, O⟩^{⊥}). Set

N(Y) := M′(Y) ⊕ M(S − Y_{red}).  (3.7)

Then N(Y) ⊂ ⟨Y ⟩^{⊥}.

Let us consider the natural projection map

π : ⟨Y ⟩^{⊥} ≃ ⟨Y, O⟩^{⊥} ⊕ ZY ⟶ ⟨Y, O⟩^{⊥}.

We claim that:

Claim : π_{|N(Y)} is injective.  (3.8)

If the claim is true, we see that N(Y) ≃ π(N(Y)) ⊂ ⟨Y, O⟩^{⊥} ≃ E_{8}^{−}. This implies that N(Y) is a negative-definite lattice generated by (−2)-elements. Hence one can see that N(Y) is a root lattice which is a direct sum of root lattices of type A_{i}, D_{j}, E_{k}. (This also implies that M′(Y) and M(S − Y_{red}) are direct sums of root lattices of type A_{i}, D_{j}, E_{k}.) To show the claim (3.8), it suffices to show that Ker π_{|N(Y)} = Ker π ∩ N(Y) = {0}. Since Ker π = Z[Y] with Y^{2} = 0 and M′(Y) is negative-definite, we have

Ker π∩N(Y) = Z[Y]∩N(Y) = Z[Y]∩(M′(Y)⊕M(S−Y_{red})) = Z[Y]∩M(S−Y_{red}).

Hence we have to show that Z[Y]∩M(S−Y_{red}) = {0}. Take γ ∈ Ker π_{|M(S−Y_{red})} and assume that γ ≠ 0. Since Ker π = Z · Y, we can write γ as γ = b · Y with b ≠ 0. We may assume that b > 0. On the other hand, since γ ∈ M(S − Y_{red}), we can write γ as

γ = C − D

with

C = ∑_{i=1}^{l} a_{i}C_{i}, D = ∑_{j=1}^{t} b_{j}D_{j}

where C_{i} (1 ≤ i ≤ l) and D_{j} (1 ≤ j ≤ t) are different (−2)-curves in S − Y_{red} and a_{i} ≥ 0, b_{j} ≥ 0. Assume that D = 0. Then we see that bY and C are linear equivalent to each other. Since bY and C are different effective divisors, we see that dim H^{0}(S, O_{S}(bY)) ≥ 2. This contradicts to the fact that (S, Y) is of non-fibered type (cf. Proposition 1.3, [STT]). Therefore we may assume that both of C and D are non-zero effective divisors. Recall that the lattice ⟨Y ⟩^{⊥} is negative semi-definite. Hence one has

0 ≥ C^{2} = (D + bY)^{2} = D^{2} = D · C ≥ 0.

(Here we used the fact that D · Y = C · Y = 0). This implies that

C^{2} = D^{2} = C · D = 0.

An element G ∈ ⟨Y ⟩^{⊥} with G^{2} = 0 must be proportional to Y, that is, G = cY. Therefore we see that C = b′Y with b′ > 0, which again contradicts to the fact that (S, Y) is of non-fibered type. We have proved that Ker π_{|M(S−Y_{red})} = {0} and hence Ker π_{|N(Y)} = {0} as in (3.8).

By Lemma 3.1, there are only finitely many (−2) curves {C_{i}}_{i=1}^{l} on S − Y_{red}. The dual graph of configurations of (−2)-curves on S can be classified by the Dynkin diagram of ADE types. The following theorem is the main theorem in this section.

Theorem 3.1. Let (S, Y) be a rational Okamoto–Painlevé pair of nonfibered type which corresponds to a Painlevé equation (cf. Table 1). The type of the root lattice M(S −Y_{red}), or equivalently, the dual graph of the configuration of (−2)-curves on S − Y are classified in Table 2.

Painlevé R(Y) the type of the dual graph of configuration of
equations  (−2)-curves on S − Y
P_{V I} D̃_{4} D_{4} , (A_{1}, A_{1}, A_{1}, A_{1}), A_{3}
(A_{1}, A_{1}, A_{1}) , A_{2}, (A_{1}, A_{1}) , A_{1}
P_{V}  D̃_{5} A_{3}, A_{2}, (A_{1}, A_{1}), A_{1}
P D̃_{6} ; III  D̃_{6} (A_{1}, A_{1}), A_{1}
P D̃_{7} ; III  D̃_{7} none
P D̃_{6} ; III  D̃_{8} none
P_{IV}  Ẽ_{6}  A_{2}, A_{1}
P_{II}  Ẽ_{7}  A_{1}
P_{I}  Ẽ_{8}  none

Table 2. Configuration of (−2)-curves on S−Y for a rational Okamoto–Painlevé pair (S, Y) of non-fibered type

3.2. The case of fibered type

Oguiso and Shioda [O-S] give the complete structure theorem of the Mordell-Weil group of rational elliptic surfaces f : S ⟶ P^{1} with a section. Let (S, Y) be a rational Okamoto–Painlevé pair of fibered type, i.e. there exists an elliptic fibration f : S → P^{1} such that f^{∗}(∞) = Y. Since K_{S} = f^{∗}(−∞) = −Y, by the adjunction formula, it is easy to check that an irreducible curve C is a (−2)-curve if and only if it is one of the irreducible components of the reducible singular fibers. Hence, to give the complete structure of (−2)-curves on S − Y, we quote the structure of the reducible singular fibers which is a part of the structure theorem of the Mordell-Weil group of f : S ⟶ P^{1}.

We will introduce some notations. Let (S, Y) be a rational Okamoto– Painlevé pairs of fibered type with an elliptic fibration f : S ⟶ P^{1} such that f^{∗}(∞) = Y. (Here, we do not assume that the type of Y is in Table 1). We also assume that there exists a section O ⊂ S and we denote by F the class of a general fiber of f so that Y and F are linearly equivalent to each other, or equivalently, have the same class in H^{2}(S, Z). For a lattice L, let us denote by L^{−} the opposite lattice of L, i.e.,

L^{−} = the module L with the pairing (−1)× < , > .

Let F_{v} := f^{−1}(v) denote the fiber over the closed point v ∈ P^{1}, and set

Sing(f) := {v ∈ P^{1}|F_{v} = f^{−1}(v) is singular },
R = Red(f) := {v ∈ P^{1}|F_{v} = f^{−1}(v) is reducible }.

For each v ∈ R, let

F_{v} = f^{−1}(v) = Θ_{v,0} + ∑_{i=1}^{m_{v}−1} μ_{v,i}Θ_{v,i} (μ_{v,i} ≥ 1, μ_{v,0} = 1)

be the irreducible decomposition of F_{v} where Θ_{v,0} is the unique component of F_{v} meeting the zero section O and m_{v} is the number of irreducible components. We set

T_{v} := ⟨Θ_{v,i}|1 ≤ i ≤ m_{v} − 1⟩_{Z} ⊂ NS(S),  (3.9)

and

T := ⨁_{v∈R} T_{v}.  (3.10)

Note that the notation T is used for another lattice in [Shi].

By the classification of singular fibers (cf. [Kod]), (and using the intersection matrix (Θ_{v,i} · Θ_{v,j})_{1≤i,j≤m_{v}−1}), we have the following

Lemma 3.2 (Lemma 7.2 [Shi]). The opposite lattice T_{v}^{−} is a root lattice of rank m_{v} − 1, determined by the type of the singular fiber F_{v} as follows:

\begin{matrix}Type of F_{v} & I_{m} & I_{m}^{∗} & II^{∗} & III^{∗} & IV^{∗} & IV & III \\ T_{v}^{−} & A_{m−1} & D_{m+4} & E_{8} & E_{7} & E_{6} & A_{2} & A_{1} \end{matrix}

Furthermore, we have (cf. (7.2) [Shi])

⟨O, F, Θ_{v,i} (0 ≤ i ≤ m_{v}−1, v ∈ R)⟩_{Z} = ⟨O, F ⟩_{Z}⊕T ⊂ NS(S) (orthogonal direct sum)

where F is the class of a fiber of f. As we see in the previous subsection, we see that ⟨O, F ⟩^{⊥} ≃ E_{8}^{−}.

Hence we have an embedding

T^{−} = ⨁_{v∈R} T_{v}^{−} ↪ E_{8}.  (3.11)

Now we recall Dynkin’s results on the classification of root lattices contained in E_{8}, which is equivalent to the classification of regular semisimple subalgebras of the exceptional Lie algebra of type E_{8}.

Theorem 3.2 (Ch. II, Table 11 [D]). Let L be a root lattice of rank s which is embedded as a sub-lattice of E_{8}, other than {0} and E_{8}. Then L is isomorphic to one in Table 3.

s L
8 A_{8}, D_{8}, A_{7} ⊕ A_{1}, A_{5} ⊕ A_{2} ⊕ A_{1}, A_{4}^{⊕2}, A_{2}^{⊕4}, E_{6} ⊕ A_{2}, E_{7} ⊕ A_{1}
D_{6} ⊕ A_{1}^{⊕2}, D_{5} ⊕ A_{3}, D_{4}^{⊕2}, D_{4} ⊕ A_{1}^{⊕4}, A_{3}^{⊕2} ⊕ A_{1}^{⊕2}, A_{1}^{⊕8}
7 A_{6} ⊕ A_{1}, A_{4} ⊕ A_{2} ⊕ A_{1}, A_{5} ⊕ A_{2}, A_{2}^{⊕3} ⊕ A_{1}, E_{6} ⊕ A_{1}, E_{7}, D_{7},
D_{5} ⊕ A_{1}^{⊕2}, D_{4} ⊕ A_{1}^{⊕3}, A_{3}^{⊕2} ⊕ A_{1}, A_{1}^{⊕7}, D_{6} ⊕ A_{1}, D_{5} ⊕ A_{2},
A_{3} ⊕ A_{2} ⊕ A_{1}^{⊕2}, D_{4} ⊕ A_{3}, A_{3} ⊕ A_{1}^{⊕4}, A_{4} ⊕ A_{3}, A_{5} ⊕ A_{1}^{⊕2}, A_{7}
6 A_{2}^{⊕3}, E_{6}, D_{6}, D_{4} ⊕ A_{1}^{⊕2}, A_{3}^{⊕2}, D_{5} ⊕ A_{1}, A_{3} ⊕ A_{1}^{⊕3}, D_{4} ⊕ A_{2},

A_{1}^{⊕6}, A_{2} ⊕ A_{1}^{⊕4}, A_{4} ⊕ A_{1}^{⊕2}, A_{6}, A_{3} ⊕ A_{2} ⊕ A_{1}, A_{5} ⊕ A_{1}, A_{4} ⊕ A_{2}

A_{2}^{⊕2} ⊕ A_{1}^{⊕2}
5 D_{5}, A_{3} ⊕ A_{1}^{⊕2}, A_{3} ⊕ A_{2}, A_{5} , A_{1}^{⊕5}, A_{4} ⊕ A_{1}, D_{4} ⊕ A_{1}
A_{2} ⊕ A_{1}^{⊕3}, A_{2}^{⊕2} ⊕ A_{1}
4 D_{4}, , A_{1}^{⊕4}, A_{2} ⊕ A_{1}^{⊕2}, A_{2}^{⊕2}, A_{3} ⊕ A_{1}, A_{4}
3 A_{3}, A_{2} ⊕ A_{1}, A_{1}^{⊕3}
2 A_{2}, A_{1}^{⊕2}
1 A_{1}

Table 3. Root sub-lattices of E_{8}

From Theorem 3.2, one can classify the root sub-lattice of E_{8}, hence T must be one of the root lattices in the Table 3.

However, as for the existence, we quote the following

Theorem 3.3 (cf. Remark 2.7 [O-S]). For every type given in Table 3 except for the type

D_{4} ⊕ A_{1}^{⊕4}, A_{1}^{⊕8} and A_{1}^{⊕7},

there exists a rational elliptic surface whose T^{−} is of given type.

Remark 5 (cf. Remark 3.4. [O-S]). The sum of the local Euler number of the reducible singular fibers cannot exceed 12, the Euler number of a rational elliptic surface. Therefore, the types D_{4} ⊕ A_{1}^{⊕4}, A_{1}^{⊕8} and A_{1}^{⊕7} do not appear.

In the case of a rational Okamoto–Painlevé pair (S, Y) of fibered type in Table 1, the type of root lattice T_{∞} is determined by the type of Y. Thus, we obtain the classification theorem as follows.

Proposition 3.1. Let (S, Y) be a rational Okamoto–Painlevé pair of fibered type in Table 1. The type of root lattice ⨁_{v∈R−∞} T_{v}^{−} are classified by Table 4.

Type of Y ⨁_{v∈R−∞} T_{v}^{−}
D̃_{4} = I_{0}^{∗}  D_{4}, A_{3}, A_{1}^{⊕3}, A_{2}, A_{1}^{⊕2}, A_{1}
D̃_{5} = I_{1}^{∗}  A_{3}, A_{2}, A_{1}^{⊕2}, A_{1}
D̃_{6} = I_{2}^{∗}  A_{1}^{⊕2}, A_{1}
D̃_{7} = I_{3}^{∗}  none
D̃_{8} = I_{4}^{∗}  none
Ẽ_{6} = IV^{∗} A_{2}, A_{1}
Ẽ_{7} = III^{∗} A_{1}
Ẽ_{8} = II^{∗} none

Table 4. The list of root lattices ⨁_{v∈R−∞} T_{v}^{−} (fibered type)

Remark 6. By Lemma 3.2 and Proposition 3.1, the structure of configuration of (−2)-curves (i.e. type of F_{v}’s) is ‘almost’ determined. For T_{v} = A_{1}, the type of F_{v} cannot be distinguished between I_{2} and III. Similarly, for T_{v} = A_{2}, the type of F_{v} cannot be distinguished between I_{2} and IV. (For other types, we can determine the type of F_{v}). In the case of Y = D̃_{4} = I_{0}^{∗}, one has the root lattice T_{v}^{−} ≃ D_{4}. In this case the corresponding fiber F_{v} is of type D̃_{4} = I_{0}^{∗}. Note that there is a sub-lattice A_{1}^{⊕4} (or the configuration of (−2)-curves (A_{1}, A_{1}, A_{1}, A_{1})) in D̃_{4}. This sub-lattice gives the configuration of type (A_{1}, A_{1}, A_{1}, A_{1}) after the deformation from fibered type to non-fibered type. (See Figure 2).

3.3. Proof of Theorem 3.1

Now we prove Theorem 3.1. Let (S, Y) be a rational Okamoto–Painlevé pair of non-fibered type with a given type R = R(Y) of Y in the Table 1. Let M′(Y) and M(S −Y_{red}) be the sub-lattices defined in (3.6). By Lemma 3.1, the orthogonal sum M′(Y)^{−} ⊕M(S −Y_{red})^{−} is a root sub-lattice of E_{8}. Then since the type R′(Y) of M′(Y)^{−} is D_{k}, 4 ≤ k ≤ 8 or E_{6}, E_{7}, E_{8}, by the Classification Theorem 3.2, we can obtain the list of possible types for M(S − Y_{red})^{−} as in Table 2. Therefore, it suffices to show for each type R′′ of root lattices listed in Table 2, there exsits a rational Okamoto–Painlevé pair (S, Y) of non-fibered types with the root sub-lattice M(S − Y_{red}) of type R′′.

First, let (S, Y) be a rational Okamoto–Painlevé pair (S, Y) of fibered type with a given type of Y in the Table 1 and let f : S ⟶ P^{1} be the elliptic fibration with f^{∗}(∞) = Y. From Proposition 3.1, we can determine the possible configuration of (−2)-curves on S − Y_{red} = S − f^{−1}(∞) by the classification of the other reducible singular fibers. (Note that Proposition 3.1 says the existence of such a fibration.) Let Y = ∑_{i=1}^{r} m_{i}Y_{i} be the irreducible decomposition of Y. Set D = Y_{red} = ∑_{i=1}^{r} Y_{i}, and take all (−2) curves {C_{1}, ⋯, C_{l}} on S −Y_{red}. Note that each C_{i} is an irreducible component of reducible singular fibers of f.

Now we will use the following deformation argument.

Lemma 3.3. Let (S, Y) be a rational Okamoto–Painlevé pair of fibered type with the irreducible decomposition Y = ∑_{i=1}^{r} m_{i}Y_{i} such that D = Y_{red} is a normal crossing divisor, and let C = ∑_{j=1}^{s} C_{j} be a normal crossing divisor of S satisfying the following conditions:

1. C ⊂ S − D,

2. C_{j} ≃ P^{1},

3. The classes of curves {Y_{i}, C_{j} | 1 ≤ i ≤ r, 1 ≤ j ≤ s} are linearly independent in H^{2}(S, C) ≃ Pic(S) ⊗_{Z} C.

Then there exists a rational Okamoto–Painlevé pair (S′, Y′) such that 1. (S′, Y′) is of non-fibered type,

2. the type of Y′ is same as the type of Y,

3. S′ − Y′_{red} contains (−2) curves {C′_{j}}_{j=1}^{s} with the same configurations as {C_{j}}_{j=1}^{s}, and

4. S′ is a deformation of S.

Proof. Let F be an arbitrary fiber at P^{1} − {∞} − Sing(f), which is an elliptic curve and F ⊂ S − (D + C). In Lemma 3.4, we will show

H^{2}(Θ_{S}(− log(D + C + F))) = {0}.  (3.12)

By (3.12), the exact sequence of sheaves

0 → Θ_{S}(− log(D + C + F)) → Θ_{S}(− log(D + C)) → N_{F} → 0,

yields the exact sequence

H^{1}(Θ_{S}(− log(D + C + F))) → H^{1}(Θ_{S}(− log(D + C))) →^{ϕ} H^{1}(N_{F}) → 0. (3.13)

Since F and Y are linearly equivalent, we get N_{F} = [F]_{|F} = [Y]_{|F} = O_{F}, and hence H^{1}(N_{F}) = H^{1}(O_{F}) = C. From (3.13) together with (3.12), we see that there exists an element θ ∈ H^{1}(Θ_{S}(− log(D +C))) such that ϕ(θ) ≠ 0. Such an element θ induces an infinitesimal deformation of the pair (S, D+C) which does not preserve the elliptic curve F. Since we see that H^{2}(S, Θ_{S}(− log(D +C))) = {0}, such an infinitesimal deformation θ induces a one parameter deformation

S ↩ D + C
φ ↓  ↙

Δ

of (S, D + C) where Δ = {z ∈ C | |z| < ϵ} is a small neighborhood of the origin. Note that we also have the relative divisor Y_{i} for φ which gives the deformation of Y_{i}. Hence we have the relative divisor Y = ∑_{i=1}^{r} m_{i}Y_{i}. For z ∈ Δ, denote by S_{z}, Y_{i,z}, D_{z}, C_{z}, Y_{z} the corresponding fibers of S, Y_{i}, D, C and Y over z respectively. It is obvious that for every z ∈ Δ each Y_{i,z} is a (−2)- curve on S_{z} and Y_{z} satisfies the numerical condition (2.1) that Y_{z} · Y_{i,z} = 0 for all i.

Consider the divisor K_{S} + Y on S and set L = O_{S}(K_{S} + Y). We know the following two facts:

1. L_{|S_{0}} ∼ O_{S_{0}}.

2. Since S_{z} is a projective smooth rational surface for every z ∈ Δ, we see that H^{i}(S_{z}, O_{S_{z}}) = 0 for i ≥ 1 and every z ∈ Δ. In particular, R^{i}π_{∗}O_{S} = 0 for i ≥ 1. Then by the upper-semicontinuity theorem, we see that dim H^{i}(S_{z}, L_{|S_{z}}) = 0 for every i ≥ 1 and z ∈ Δ. Noting that π_{∗}L ≃ O_{Δ}, we see that there is a nontrivial homomorphism s : π^{∗}(π_{∗}L) = O_{S} → L. Applying the same argument for the dual sheaf L^{∨}, we also have a non-trivial homomorphism s′ : O_{S} → L^{∨}. Then we conclude that O_{S}(K_{S}+Y) = L ≃ O_{S}. Therefore we see that K_{S} = −Y and hence K_{S_{z}} ∼ −Y_{z} for every z ∈ Δ. This implies that (S_{z}, Y_{z}) is a rational Okamoto–Painlevé pair for z ∈ Δ. Next we claim that there exists z ∈ Δ − {0} such that dim H^{0}(S_{z}, Y_{z}) = 1. This also implies that (S_{z}, Y_{z}) is of non-fibered type. If dim H^{0}(S_{z}, Y_{z}) ≥ 2 for every z ∈ Δ, we can show that there exists an elliptic fibration f_{z} : S_{z} ⟶ P^{1} with f_{z}^{∗}(∞) = Y_{z} which is a deformation of the original elliptic fibration f : S_{0} ⟶ P^{1}. Since the general fiber F of f does not extend over z ∈ Δ − {0}, this deduces the contradiction. Note that the type of (S_{z}, Y_{z}) is same as the type of (S_{0}, Y_{0}) = (S, Y) and S_{z} − (Y_{z})_{red} contains (−2)-curves C_{z} whose configuration is same as the configuration of C_{0} = C. .

Now we shall prove the claim (3.12).

Lemma 3.4. Under the same assumption of Lemma 3.3, we have

H^{2}(S, Θ_{S}(− log(D + C + F)) = {0},

where F is a smooth fiber of the elliptic fibration f : S ⟶ P^{1}.

Proof. By the Serre duality, it suffices to show that

H^{0}(S, Ω_{S}^{1}(log(D + C + F)) ⊗ K_{S}) ≃ H^{0}(S, Ω_{S}^{1}(log(D + C + F))(−F)) = {0}. (3.14)

(Note that K_{S} ∼ −F). Set D̃ = ∏_{i=1}^{r} Y_{i}, C̃ = ∏_{j=1}^{s} C_{j}. Then we have the following commutative diagram of sheaves:

0 ↓

0  0  →  O_{F} (−F)  →^{δ}
↓  ↓  ↓
0 →  Ω_{S}^{1}(−F)  →  Ω_{S}^{1}  →  (Ω_{S}^{1})_{|F}  → 0
↓  ↓  ↓ μ
0 → Ω_{S}^{1}(log(D + C + F))(−F) → Ω_{S}^{1}(log(D + C + F)) → Ω_{S}^{1}(log(F))_{|F} → 0
↓ P.R.  ↓ P.R.  ↓
→^{δ}  ⊕O_{Y_{i}} ⊕ O_{C_{j}} ⊕ O_{F} (−F) → ⊕O_{Y_{i}} ⊕ O_{C_{j}} ⊕ O_{F} →  O_{F}  → 0
↓  ↓  ↓
0  0  0  .

(3.15)

Here the map P.R. : Ω_{S}^{1}(log(D+C +F)) ⟶ ⊕_{i=1}^{r}O_{Y_{i}} ⊕_{j=1}^{s} O_{C_{j}} is the Poincaré residue map and the image of μ : (Ω_{S}^{1})_{|F} ⟶ Ω_{S}^{1}(log F)_{|F} coincides with Ω_{F}^{1} so that the following sequences are exact.

0 ⟶ O_{F} (−F) ⟶ (Ω_{S}^{1})_{|F} ⟶ Ω_{F}^{1} ⟶ 0,  (3.16)
0 ⟶ Ω_{F}^{1} ⟶ (Ω_{S}^{1}(log F))_{|F} ⟶ O_{F} ⟶ 0.  (3.17)

Noting that N_{F}^{∨} ≃ O_{F} (−F) ≃ O_{F} and H^{0}(Ω_{S}^{1}) = 0, from the first and second rows of (3.15), we obtain the exact sequence of cohomology

\begin{aligned}& 0 \\ & ↓ \\ & H^{0}(O_{F} (−F)) ≃ C \\ & ↓ H_{1} & ↘ \\ 0 ⟶ & H^{0}((Ω_{S}^{1})_{|F}) & → & H^{1}(Ω_{S}^{1}(−F)) \end{aligned}  (3.18)

From the first column of (3.15), H^{0}(Ω_{S}^{1}(log(D + C + F))(−F)) is isomorphic to the kernel of Gysin map

⊕_{i=1}^{r}H^{0}(O_{Y_{i}}) ⊕_{j=1}^{s} H^{0}(O_{C_{j}}) ⊕ H^{0}(O_{F} (−F)) ⟶^{G_{1}} H^{1}(Ω_{S}^{1}(−F)),  (3.19)

We will show that the Gysin map G_{1} is injective, which implies the assertion (3.14).

By (3.15) and (3.18), we can decompose the map G_{1} as follows:

\begin{matrix}0 & & 0 \\ ↓ & & ↓ \\ H^{0}(O_{F} (−F)) ≃ C & ⟶^{H_{1}} & H^{0}((Ω_{S}^{1})_{|F}) \\ ↓ τ & & ↓ \\ \begin{matrix}H^{0}(O_{F} (−F)) ≃ C \\ ⊕ \\ ⊕_{i=1}^{r}H^{0}(O_{Y_{i}}) ⊕_{j=1}^{s} H^{0}(O_{C_{j}}) \end{matrix} \} & ⟶^{G_{1}} & H^{1}(Ω_{S}^{1}(−F)) \\ ↓ μ_{1} & & ↓ ν \\ ⊕_{i=1}^{r}H^{0}(O_{Y_{i}}) ⊕_{j=1}^{s} H^{0}(O_{C_{j}}) & ⟶^{G_{2}} & H^{1}(Ω_{S}^{1}). \end{matrix}  (3.20)

Here μ_{1} is just the projection and G_{2} is the natural Gysin map. Since H_{1} is injective (cf. (3.18)), a diagram chasing shows that G_{1} is injective if G_{2} is injective. The image of 1_{Y_{i}} and 1_{C_{j}} by G_{2} are the class of the divisors of Y_{i} and C_{j} in H^{1}(Ω_{S}^{1}) ≃ H^{2}(S, C). Since {Y_{i}, C_{j}, 1 ≤ i ≤ r, 1 ≤ j ≤ s} are linearly independent in H^{2}(S, C) ≃ H^{1}(Ω_{S}^{1}) by assumption of Lemma 3.3, G_{2} is injective, hence we have proved the assertion. .

Now together with Lemma 3.3 and Proposition 3.1 the following lemma shows the existence part of Theorem 3.1 and hence completes the proof of Theorem 3.1. (See Example 3.1).

Lemma 3.5. Let R be a type of affine root lattice in Table 1, that is R = Ẽ_{k}, (k = 8, 7, 6) or R = D̃_{l}, (l = 8, 7, 6, 5, 4). Let (S, Y) be a rational Okamoto– Painlevé pair of fibered-type and let f : S ⟶ P^{1} be the elliptic fibration with f^{∗}(∞) = Y = ∑_{i=1}^{r} m_{i}Y_{i}. Let {C_{j}}_{j=1}^{s} be a set of different irreducible (−2) curves on S − Y_{red} such that no linear combination of {C_{j}}_{j=1}^{s} has the same class of general fiber (= the class of Y). Then { Y_{i}, C_{j} | 1 ≤ i ≤ r, 1 ≤ j ≤ s} are linearly independent in H^{2}(S, Q).

Proof. From the condition of the set {C_{j}}_{j=1}^{s}, we see that the sub-lattice ⟨C_{j}⟩_{j=1}^{s} ⊂ H^{2}(S, Z) generated by {C_{j}}_{j=1}^{s} is negative-definite. Then we have an orthogonal decomposition

⟨C_{j}⟩_{j=1}^{s} ⊕ ⟨Y_{i}⟩_{i=1}^{r} ⊂ H^{2}(S, Z)

which shows the assertion.

Example 3.1. From Proposition 3.1, we have an Okamoto-Painlevé pair (S, Y) of fibered type with another singular fiber F_{1} where the pair (Y, F_{1}) has the type (D̃_{4}, D̃_{4}). Take a proper subset {C_{j}}_{j=1}^{s} of all of irreducible components of F_{1}. Then the type of M_{1} coincides with the proper subgraph of the Dynkin diagram D̃_{4} of F_{1}, that is, one of the types; D_{4}, (A_{1}, A_{1}, A_{1}, A_{1}) A_{3}, (A_{1}, A_{1}, A_{1}), A_{2}, (A_{1}, A_{1}) and A_{1}. It is easy to see that the set of classes {Y_{i}, C_{j}, 1 ≤ i ≤ 5, 1 ≤ j ≤ s} are linearly independent in H^{2}(S, Q). Therefore from Lemma 3.3 and Lemma 3.5, we see that there exists a rational Okamoto– Painlevé pair (S′, Y′) of non-fibered type, such that:

1. the type of Y′ is D̃_{4},

2. there exist (−2)-curves {C′_{j}}_{j=1}^{s} on S′ − Y′_{red} with the same Dynkin type of {C_{j}}_{j=1}^{s}. Therefore, we can obtain the assertion of Theorem 3.1 for D̃_{4}. We can treat the other cases similarly.

4. Non-existence of Riccati solutions for P_{I}, P D̃_{8} ; III, P D̃_{7} ; III

As a corollary to Theorem 3.1, we obtain the following

Corollary 4.1. Let (S, Y) be a rational Okamoto–Painlevé pair of nonfibered type, with the type R = R(Y) = Ẽ_{8}, D̃_{8} or D̃_{7}. Then S − Y_{red} does not contain a rational nodal curve C. Therefore all the Painlevé equations of types P_{I}, P D̃_{8} ; III, P D̃_{7} ; III do not admit Riccati solutions.

Proof. The first assertion directly follows from Theorem 3.1 and the last assertion follows from the first and Proposition 2.1. .

Remark 7.

1. Umemura proved that the Painlevé equation of type P_{I} has no classical solution and hence in particular no Riccati solution (cf. [Ni], [U1], [U2]).

2. Ohyama [Ohy] showed that all the Painlevé equations of type P D̃_{7} ; III has no Riccati solutions by proving that they have no invariant divisor with respect to the vector field (2.4).

3. It is worth while remarking that the obstruction to the existence of nodal curves in S − Y_{red} is a topological one and hence so is the obstruction to the existence of Riccati solutions. In fact, the sub-lattice M(S − Y_{red}) is classified only by the intersection theory of the surface S and the structure of the sub-lattice does not depend on the complex structure of S.

For other types R, by the similar argument in the proof of Lemma (A.2), we can show the following proposition. This proposition shows that for a general parameter α ∈ M_{R}, the corresponding Painlevé equations do not admit any Riccati solution.

Proposition 4.1. Let (S, Y) be a rational Okamoto–Painlevé pair of non-fibered type and of type R which corresponds to a Painlevé equation and assume that S−Y_{red} contains a nodal curve C. Then there exists a one parameter deformation of Okamoto–Painlevé pairs of non-fibered type and of the given type R, Y ↪ S ⟶ Δ = {z ∈ C||z| < ϵ} of (S, Y) such that S_{z} − Y_{z} does not contains any nodal curve for z ∈ Δ − {0}. Hence for z ∈ Δ − {0} the Painlevé equation corresponding to (S_{z}, Y_{z}) does not admit any Riccati solutions

5. Examples of (−2)-curves on S − D

In this section, we will give examples of (−2)-curves C on S − D for some rational Okamoto–Painlevé pairs (S, Y) and Riccati equations associated to C.

Here we will use the explicit description of families of Okamoto–Painlevé pairs

\begin{matrix}S & ↩ & D \\ π ↓ & ↙ & φ \\ M_{R} × B_{R} \end{matrix}  (5.1)

in [T2]. As we explained in Section 2, we have isomorphisms M_{R} = Spec M_{R} and B_{R} = Spec B_{R} such that Spec M_{R} and Spec B_{R} are affine open subschemes of Spec C[α_{1}, ⋯, α_{s}] ≃ C^{s} and Spec C[t] respectively. Moreover S can be covered by affine open sets {Ũ_{i}}_{i=1}^{l+k} such that for each i

Ũ_{i} ≃ Spec ((M_{R} ⊗ B_{R})[x_{i}, y_{i}, \frac{1}{f_{i}(x_{i},y_{i},α,t)}]) ⊂ Spec C[α, t, x_{i}, y_{i}] ≃ C^{s+3}

|| C^{12−r} (5.2)

where f_{i}(x_{i}, y_{i}, α, t) is an element of (M_{R} ⊗ B_{R})[x_{i}, y_{i}]. ( Note that in most cases f_{i}(x_{i}, y_{i}, α, t) ≡ 1 and we may assume that S − D is covered by {Ũ_{i}}_{i=1}^{l}.)

For a given point α = (α_{1}, ⋯, α_{s}) ∈ M_{R}, we denote the restriction of the family π : S ⟶ M_{R} × B_{R} to {α} × B_{R} by S_{α} ⟶ {α} × B_{R}. Moreover we set

U_{i}_{α} := Ũ_{i} ∩ S_{α} ⊂ Spec B_{R}[x_{i}, y_{i}], U_{i}_{(α,t)} = Ũ_{i} ∩ S_{α,t} ⊂ Spec C[x_{i}, y_{i}] (5.3)

where S_{α,t} = π^{−1}((α, t)).

Next let us consider the smooth variety obtained by patching affine planes W_{i} = Spec C[x_{i}, y_{i}] ≃ C^{2} ( i = 1, 2) by the coordinate transformation

x_{1} = \frac{1}{x_{2}} , y_{1} = x_{2}^{2}y_{2}.  (5.4)

It is easy to see that the equations {y_{1} = y_{2} = 0} define a (−2)-curve C in W.

Example 5.1 (Ẽ_{7}–type (P_{II})). In the case of R = Ẽ_{7} (P_{II}), the family is constructed as follows (cf. [MMT], [T2], [SU]). Let us set

M_{R} = Spec C[α] ≃ C, B_{R} = Spec C[t] ≃ C.

Here we only give the affine covering of the family π : S − D ⟶ M_{R} × B_{R}. Take three affine schemes i = 0, 1, 2

Ũ_{i} = Spec C[α, t, x_{i}, y_{i}] ≃ C^{4},  (5.5)

and patch these affine schemes by the coordinate transformations:

\begin{aligned}x_{0} & = \frac{1}{x_{1}} & = \frac{1}{x_{2}} , \\ y_{0} & = x_{1}((−α − \frac{1}{2}) − x_{1}y_{1}) & = 2x_{2}^{−2} + t + (α − \frac{1}{2})x_{2} − y_{2}x_{2}^{2}. \end{aligned}  (5.6)

On Ũ_{0}, the Painlevé vector field ṽ in (2.4) is explicitly given by

ṽ = \frac{∂}{∂t} + [y_{0} − x_{0}^{2} − \frac{t}{2}] \frac{∂}{∂x_{0}} + [2x_{0}y_{0} + α + \frac{1}{2}] \frac{∂}{∂y_{0}} .  (5.7)

which is equivalent to the equation:

\{ \begin{matrix}\frac{dx_{0}}{dt} & = & y_{0} − x_{0}^{2} − \frac{t}{2} \\ \frac{dy_{0}}{dt} & = & 2x_{0}y_{0} + α + \frac{1}{2} \end{matrix}  (5.8)

Then for α = −\frac{1}{2}, on U_{0,− \frac{1}{2}} ∪ U_{1,− \frac{1}{2}} , we obtain a family of (−2)-curves C_{− \frac{1}{2}} ⟶ \{−\frac{1}{2}\} × BẼ_{7} defined by

C_{− \frac{1}{2}} = \{y_{0} = y_{1} = 0\} ⊂ U_{0,− \frac{1}{2}} ∪ U_{1,− \frac{1}{2}} ⊂ S_{− \frac{1}{2}} − D_{− \frac{1}{2}} .  (5.9)

Moreover, on the family C_{− \frac{1}{2}} ⟶ \{−\frac{1}{2}\}×BẼ_{7}, the equation (5.8) can be reduced to

\frac{dx_{0}}{dt} = −x_{0}^{2} − \frac{t}{2}.  (5.10)

It is known that Bäcklund transformations give isomorphisms between S_{α} and S_{α±1}. Hence for α ∈ −\frac{1}{2} + Z , the family S_{α} − D_{α} also contains a family of (-2)-curves (cf. [SU], [U-W1]). Moreover, Noumi and Okamoto [NO] proved the following Theorem (cf. [Theorem 2, [NO]] and remark after it). (See also [Theorem 2,1, [U-W1]]).

Theorem 5.1. ( [Theorem 2, [NO]]). Let us denote by P_{II}(α) the equation in ( 5.8). Then

1. For every integer α ∈ Z, there exists a unique rational solution of the system P_{II}(α).

2. For every α ∈ \frac{1}{2} + Z, there exists a unique one parameter family of classical solutions of P_{II}(α), of which each solution is rationally written by a solution of the Riccati equation ( 5.10).

3. Let (x_{0}, y_{0}) be a solution of P_{II}(α) different from those mentioned above. Then neither x_{0} nor y_{0} is classical, hence a solution of a Riccati equation.

Note that for α = 0, P_{II}(0) in (5.8) has a rational solution (x_{0}, y_{0}) = (0, \frac{t}{2}). Theorem 5.1 says that this rational solution is the unique rational solution for P_{II}(0).

Example 5.2 (D̃_{4} (P_{V I})). Next let us show examples of (−2)-curves for R = D̃_{4} (cf. [T2]). The parameter space of the semiuniversal family S − D ⟶ MD̃_{4} × BD̃_{4} are given by

M_{R} = Spec C[κ_{0}, κ_{1}, κ_{∞}, κ_{t}] ≃ C^{4}, B_{R} = Spec C[t, 1/t, 1/(t−1)] ≃ C−{0, 1}.

(Here we use the parameters κ_{i}, i = 0, 1, ∞, t for M_{R} as in [MMT] and [T2].) Take affine schemes i = 0, 1, 2, 3, 4, 5

Ũ_{i} = Spec C[x_{i}, y_{i}, κ_{0}, κ_{1}, κ_{∞}, κ_{t}, t, 1/t, 1/(t − 1)] ≃ C^{2} × M_{R} × B_{R}. (5.11)

and patch them by the coordinate transformations:

\begin{matrix}x_{0} = y_{1}(κ_{0} − x_{1}y_{1}), & y_{0} = \frac{1}{y_{1}} , \\ x_{1} = y_{0}(κ_{0} − x_{0}y_{0}), & y_{1} = \frac{1}{y_{0}} , \\ x_{0} = 1 + y_{2}(κ_{1} − x_{2}y_{2}), & y_{0} = \frac{1}{y_{2}} , \\ x_{2} = y_{0}(κ_{1} + y_{0} − x_{0}y_{0}), & y_{2} = \frac{1}{y_{0}} , \\ x_{0} = t + y_{3}(κ_{t} − x_{3}y_{3}), & y_{0} = \frac{1}{y_{3}} , \\ x_{3} = y_{0}(κ_{t} + ty_{0} − x_{0}y_{0}), & y_{3} = \frac{1}{y_{0}} , \\ x_{0} = \frac{1}{x_{4}} , & y_{0} = x_{4}(\frac{κ_{0} + κ_{1} + κ_{t} − 1 + κ_{∞}}{2} − x_{4}y_{4}), \\ x_{4} = \frac{1}{x_{0}} , & y_{4} = x_{0}(\frac{κ_{0} + κ_{1} + κ_{t} − 1 + κ_{∞}}{2} − x_{0}y_{0}), \\ x_{4} = y_{5}(κ_{∞} − x_{5}y_{5}), & y_{4} = \frac{1}{y_{5}} , \\ x_{5} = y_{4}(κ_{∞} − x_{4}y_{4}), & y_{5} = \frac{1}{y_{4}} \end{matrix} (5.12)

On Ũ_{0}, the Painlevé vector field ṽ in (2.4) is given by

ṽ = \frac{∂}{∂t} + A(x, y, t)\frac{∂}{∂x_{0}} + B(x, y, t)\frac{∂}{∂y_{0}} ,  (5.13)

where

A(x, y, t) := \frac{x_{0}(x_{0} − 1)(x_{0} − t)}{t(t − 1)} [2y_{0} − (\frac{κ_{0}}{x_{0}} + \frac{κ_{1}}{(x_{0} − 1)} + \frac{(κ_{t} − 1)}{(x_{0} − t)})],
B(x, y, t) := −\frac{1}{t(t − 1)} [(3x_{0}^{2} − 2(t + 1)x_{0} + t)y_{0}^{2}
−(2(κ_{0} + κ_{1} + κ_{t} − 1)x_{0} − (κ_{0} + κ_{1})t − κ_{0} − κ_{t} + 1)y_{0}
+\frac{(κ_{0}+κ_{1}+κ_{t}−1)^{2}−κ_{∞}^{2}}{4} ] .

This is equivalent to the equation:

\{ \begin{matrix}\frac{dx_{0}}{dt} & = & A(x, y, t) \\ \frac{dy_{0}}{dt} & = & B(x, y, t). \end{matrix}  (5.14)

Let us set the hyperplanes of the parameter space MD̃_{4} × BD̃_{4} as follows:

\begin{matrix}H_{0} = \{κ_{0} = 0\}, H_{1} = \{κ_{1} = 0\}, H_{t} = \{κ_{t} = 0\}, \\ H_{ϵ} = \{κ_{0} + κ_{1} + κ_{t} + κ_{∞} − 1 = 0\}, H_{∞} = \{κ_{∞} = 0\}. \end{matrix}  (5.15)

Note that each hyperplane H_{i} is a direct product of H′_{i} ⊂ MD̃_{4} and BD̃_{4}, i.e., H_{i} = H′_{i}×BD̃_{4}. Remark also that each hyperplane is one of the reflection hyperplanes of the affine Weyl group W (D̃_{4}) generated by Bäcklund transformations (cf. [NTY]).

We consider the deformation

π^{∗}(H_{0}) ⊂  S − D
π ↓  π ↓
H_{0} ⊂ MD̃_{4} × BD̃_{4}

which is given by restricting the parameter space MD̃_{4} × BD̃_{4} to H_{0}. For the subfamily (S − D)_{(0,κ_{1},κ_{t},κ_{∞})} over H_{0}, the coordinate transformation between U_{0}_{(0,κ_{1},κ_{t},κ_{∞})} and U_{1}_{(0,κ_{1},κ_{t},κ_{∞})} is given by

x_{0} = −x_{1}y_{1}^{2}, y_{0} = \frac{1}{y_{1}} .

Therefore

C_{0,(0,κ_{1},κ_{t},κ_{∞})} := {x_{0} = x_{1} = 0}

determines a family of (−2)-curves

\begin{matrix}C_{0,(0,κ_{1},κ_{t},κ_{∞})} & ↪ & (S − D)_{(0,κ_{1},κ_{t},κ_{∞})} \\ ↓ & ↙ \\ H_{0} = H′_{0} × B_{R}. \end{matrix}  (5.16)

In the same way, we obtain a family of (−2)-curves over each hyperplane H_{i} as follows:

\begin{aligned}H_{0} & : & C_{0,(0,κ_{1},κ_{t},κ_{∞})} := \{x_{0} = x_{1} = 0\} & ⊂ (S − D)_{(0,κ_{1},κ_{t},κ_{∞})} \\ H_{1} & : & C_{1,(κ_{0},0,κ_{t},κ_{∞})} := \{x_{0} = 1, x_{2} = 0\} & ⊂ (S − D)_{(κ_{0},0,κ_{t},κ_{∞})} \\ H_{t} & : & C_{t,(κ_{0},κ_{1},0,κ_{∞})} := \{x_{0} = t, x_{3} = 0\} & ⊂ (S − D)_{(κ_{0},κ_{1},0,κ_{∞})} \\ H_{ϵ} & : & C_{ϵ,(κ_{0},κ_{1},κ_{t},κ_{∞})} := \{y_{0} = y_{4} = 0\} & ⊂ (S − D)_{(κ_{0},κ_{1},κ_{t},κ_{∞})} \\ & & & (κ_{0} + κ_{1} + κ_{t} + κ_{∞} = 1) \\ H_{∞} & : & C_{∞,(κ_{0},κ_{1},κ_{t},0)} := \{x_{4} = x_{5} = 0\} & ⊂ (S − D)_{(κ_{0},κ_{1},κ_{t},0)} \end{aligned} (5.17)

By restricting the (extended) Hamiltonian system to each C_{j}, we obtain the following Riccati equation.

• On C_{0,(0,κ_{1},κ_{t},κ_{∞})} ∩ U_{0}_{(0,κ_{1},κ_{t},κ_{∞})}:

x_{0} ≡ 0, \frac{dy_{0}}{dt} = −\frac{1}{t(t − 1)}(ty_{0}^{2} + (κ_{1}t + κ_{t} − 1)y_{0} + \frac{(κ_{1} + κ_{t} − 1)^{2} − κ_{∞}^{2}}{4} ).

• On C_{1,(κ_{0},0,κ_{t},κ_{∞})} ∩ U_{0}_{(κ_{0},0,κ_{t},κ_{∞})}:

\{ \begin{matrix}x_{0} ≡ 1, \\ \frac{dy_{0}}{dt} = −\frac{1}{t(t − 1)}((1 − t)y_{0}^{2} − ((κ_{0} + κ_{t} − 1) − κ_{0}t)y_{0} + \frac{(κ_{0} + κ_{t} − 1)^{2} − κ_{∞}^{2}}{4} ). \end{matrix}

fiber  (−2)-curves  configuration
(S − D)_{(0,0,0,1)} {C_{0}, C_{1}, C_{t}, C_{ϵ}}  D_{4}
(S − D)_{(0,0,1,0)} {C_{0}, C_{1}, C_{ϵ}, C_{∞}}  D_{4}
(S − D)_{(0,1,0,0)} {C_{0}, C_{t}, C_{ϵ}, C_{∞}}  D_{4}
(S − D)_{(1,0,0,0)} {C_{1}, C_{t}, C_{ϵ}, C_{∞}}  D_{4}
(S − D)_{(0,0,0,0)} {C_{0}, C_{1}, C_{t}, C_{∞}} A_{1}, A_{1}, A_{1}, A_{1}

Table 5.

• On C_{t,(κ_{0},κ_{1},0,κ_{∞},t)} ∩ U_{0}_{(κ_{0},κ_{1},0,κ_{∞},t)}:

\{ \begin{matrix}x_{0} ≡ t, \\ \frac{dy_{0}}{dt} = −\frac{1}{t(t−1)}(t(t − 1)y_{0}^{2} − ((κ_{0} + κ_{1} − 2)t − κ_{0} + 1)y_{0} + \frac{(κ_{0}+κ_{1}−1)^{2}−κ_{∞}^{2}}{4} ). \end{matrix}

• On C_{ϵ,(κ_{0},κ_{1},κ_{t},1−(κ_{0}+κ_{1}+κ_{t}),t)} ∩ U_{0}_{(κ_{0},κ_{1},κ_{t},1−(κ_{0}+κ_{1}+κ_{t}),t)}:

\{ \begin{matrix}\frac{dx_{0}}{dt} = −\frac{1}{t(t−1)}(κ_{0}(x_{0} − 1)(x_{0} − t) + κ_{1}x_{0}(x_{0} − t) + (κ_{t} − 1)x_{0}(x_{0} − 1)), \\ y_{0} ≡ 0. \end{matrix}
• On C_{∞,(κ_{0},κ_{1},κ_{t},0,t)} ∩ U_{4}_{(κ_{0},κ_{1},κ_{t},0,t)}:

x_{4} ≡ 0, \frac{dy_{4}}{dt} = −\frac{1}{t(t − 1)}(y_{4}^{2} + ((κ_{t} − 1)t + κ_{1})y_{4} + \frac{(κ_{1} + κ_{t} − 1)^{2} − κ_{0}^{2}}{4} t).

Next, choose four hyperplanes from the five hyperplanes and consider the fibers over the intersection of them. For each fiber (S − D)_{(0,0,0,1,t)} of (0, 0, 0, 1, t) ∈ H_{0} ∩ H_{1} ∩ H_{t} ∩ H_{ϵ}, we can see that C_{0,(0,0,0,1,t)}, C_{1,(0,0,0,1,t)}, and C_{t,(0,0,0,1,t)} do not intersect each other but they intersect with C_{ϵ,(0,0,0,1,t)} respectively. Hence the type of the configuration of these curves is D_{4}. By checking the other cases, we obtain the following.

Below, we only give the tables for D̃_{k}, k = 5, 6. The case Ẽ_{6} will be treated in Section 6. For parameters and the coordinate transformations, see [T2].

Example 5.3 (D̃_{5} (P_{V})).

MD̃_{5} = Spec C[κ_{0}, κ_{t}, κ_{∞}] ≃ C^{3}, BD̃_{5} = Spec C[t, t^{−1}] ≃ C^{×}.

H_{0} = {κ_{0} = 0}, H_{ϵ} = {κ_{0} + κ_{t} + κ_{∞} = 0}, H_{∞} = {κ_{∞} = 0}.

\begin{aligned}H_{0} & : & C_{0,(0,κ_{t},κ_{∞})} := \{x_{0} = x_{1} = 0\} & ⊂ (S − D)_{(0,κ_{t},κ_{∞})} \\ H_{ϵ} & : & C_{ϵ,(κ_{0},κ_{t},−(κ_{0}+κ_{t}))} := \{y_{0} = y_{3} = 0\} & ⊂ (S − D)_{(κ_{0},κ_{t},−(κ_{0}+κ_{t}))} \\ H_{∞} & : & C_{∞,(κ_{0},κ_{t},0)} := \{x_{3} = x_{4} = 0\} & ⊂ (S − D)_{(κ_{0},κ_{t},0)} \end{aligned}

Nodal curves and Riccati solutions 29

C_{ϵ} C_{0}  C_{1}  C_{t}  C_{∞}  C_{ϵ} C_{0}  C_{1}  C_{t}  C_{∞} C_{ϵ} C_{0}  C_{1}  C_{t}  C_{∞}  C_{ϵ} C_{0}  C_{1}  C_{t}  C_{∞} C_{ϵ} C_{0}  C_{1}  C_{t}  C_{∞}

Figure 2. Maximal configurations for R = D̃_{4}.

• On C_{0,(0,κ_{t},κ_{∞})} ∩ U_{0}_{(0,κ_{t},κ_{∞})}

\{ \begin{matrix}x_{0} ≡ 0 \\ \frac{dy_{0}}{dt} = −\frac{1}{t}(y_{0}^{2} + (κ_{t} − t)y_{0} + \frac{κ_{t}^{2}−κ_{∞}^{2}}{4} ). \end{matrix}

• On C_{ϵ,(κ_{0},κ_{t},−(κ_{0}+κ_{t}))} ∩ U_{0}_{(κ_{0},κ_{t},−(κ_{0}+κ_{t}))}

\frac{dx_{0}}{dt} = −\frac{1}{t}(κ_{0}(x_{0} − 1)^{2} + κ_{t}x_{0}(x_{0} − 1) + tx_{0}), y_{0} ≡ 0.

• On C_{∞,(κ_{0},κ_{t},0)} ∩ U_{3}_{(κ_{0},κ_{t},0)}

x_{3} ≡ 0, \frac{dy_{3}}{dt} = −\frac{1}{t}(y_{3}^{2} + (κ_{t} + t)y_{3} + \frac{κ_{t}^{2} − κ_{0}^{2}}{4} ).

fiber  (−2)-curves configuration
(S − D)_{(0,0,0)} {C_{0}, Cϵ, C_{∞}}  A_{3}

Example 5.4 (D̃_{6} (P_{III})).

M_{R} = Spec C[κ_{0}, κ_{∞}] ≃ C^{2}, B_{R} = Spec C[t, t^{−1}] ≃ C^{×}.

H_{1} = {κ_{0} + κ_{∞} = 0}, H_{2} = {κ_{0} − κ_{∞} = 0},
H_{3} = {κ_{0} − κ_{∞} + 2 = 0}, H_{4} = {κ_{0} + κ_{∞} + 2 = 0}.

\begin{aligned}H_{1} & : & C_{1,(κ_{0},−κ_{0})} := \{y_{0} = y_{2} = 0\} & ⊂ (S − D)_{(κ_{0},−κ_{0})} \\ H_{2} & : & C_{2,(κ_{0},κ_{0})} := \{y_{0} = t, y_{3} = 0\} & ⊂ (S − D)_{(κ_{0},κ_{0})} \\ H_{3} & : & C_{3,(κ_{0},κ_{0}+2)} := \{y_{1} = 0, y_{2} = t\} & ⊂ (S − D)_{(κ_{0},κ_{0}+2)} \\ H_{4} & : & C_{4,(κ_{0},−κ_{0}−2)} := \{y_{1} = t, y_{3} = t\} & ⊂ (S − D)_{(κ_{0},−κ_{0}−2)} \end{aligned}

• On C_{1,(κ_{0},−κ_{0})} ∩ U_{0}_{(κ_{0},−κ_{0})}

\frac{dx_{0}}{dt} = \frac{1}{t}(−2tx_{0}^{2} − (2κ_{0} + 1)x_{0} + 2t), y_{0} ≡ 0.

• On C_{2,(κ_{0},κ_{0})} ∩ U_{0}_{(κ_{0},κ_{0})}

\frac{dx_{0}}{dt} = \frac{1}{t}(2tx_{0}^{2} − (2κ_{0} + 1)x_{0} + 2t), y_{0} ≡ t.

• On C_{3,(κ_{0},κ_{0}+2)} ∩ U_{1}_{(κ_{0},κ_{0}+2)}

\frac{dx_{1}}{dt} = \frac{1}{t}(−2tx_{1}^{2} + (2κ_{0} + 3)x_{1} − 2t), y_{1} ≡ 0.

• On C_{4,(κ_{0},−κ_{0}−2)} ∩ U_{1}_{(κ_{0},−κ_{0}−2)}

\frac{dx_{1}}{dt} = \frac{1}{t}(2tx_{1}^{2} + (2κ_{0} + 3)x_{1} − 2t), y_{1} ≡ 1.

fiber  (−2)-curves configuration
(S − D)_{(0,0)}  {C_{1}, C_{2}}  A_{1}, A_{1}
(S − D)_{(−1,1)}  {C_{1}, C_{3}}  A_{1}, A_{1}
(S − D)_{(−1,−1)}  {C_{2}, C_{4}}  A_{1}, A_{1}
(S − D)_{(−2,0)}  {C_{3}, C_{4}}  A_{1}, A_{1}

6. Confluences of Nodal Curves and Riccati Equations

In this section, we will discuss the confluence of nodal curves and Riccati equations for Painlevé equations. We will deal with only the case R = Ẽ_{6} (P_{IV}), however one can easily extend the result to other cases like D̃_{5} and D̃_{4}.

6.1. The confluence of nodal curves Example 6.1 (Ẽ_{6} (P_{IV})).

M_{R} = Spec C[κ_{0}, κ_{∞}] ≃ C^{2}, B_{R} = Spec C[t] ≃ C.

An open covering of S − D is given by

S − D = ⋃_{i=0}^{3} Ũ_{i}

where for i = 0, 1, 2, 3

Ũ_{i} = Spec C[x_{i}, y_{i}, κ_{0}, κ_{∞}, t] ≃ C^{5}.

Moreover the coordinate transformations are given by

\begin{matrix}x_{0} = y_{1}(κ_{0} − x_{1}y_{1}), & y_{0} = \frac{1}{y_{1}} , \\ x_{1} = y_{0}(κ_{0} − x_{0}y_{0}), & y_{1} = \frac{1}{y_{0}} , \\ x_{0} = \frac{1}{x_{2}} , & y_{0} = x_{2}(κ_{∞} − x_{2}y_{2}), \\ x_{2} = \frac{1}{x_{0}} , & y_{2} = x_{0}(κ_{∞} − x_{0}y_{0}), \\ x_{2} = x_{3}, & y_{2} = −\frac{1/2}{x_{3}^{3}} − \frac{t}{x_{3}^{2}} + \frac{2κ_{∞} − κ_{0} + 1}{x_{3}} + y_{3}, \\ x_{3} = x_{2}, & y_{3} = \frac{1/2}{x_{2}^{3}} + \frac{t}{x_{2}^{2}} − \frac{2κ_{∞} − κ_{0} + 1}{x_{2}} + y_{2}. \end{matrix}

Finally, on the affine open set Ũ_{0}, the Painlevé system of type Ẽ_{6} which is equivalent to P_{IV} is given as follows.

\{ \begin{matrix}\frac{dx_{0}}{dt} & = & 4x_{0}y_{0} − x_{0}^{2} − 2tx_{0} − 2κ_{0} \\ \frac{dy_{0}}{dt} & = & −2y_{0}^{2} + 2(x_{0} + t)y_{0} − κ_{∞} \end{matrix} .  (6.1)

We have two hyperplanes H_{0} and H_{∞} on M_{R} × B_{R} and families of (−2)- curves C_{0} and C_{∞} over H_{0} and H_{∞} as follows.

\begin{aligned}H_{0} = \{κ_{0} = 0\} & : & C_{0,(0,κ_{∞})} := \{x_{0} = x_{1} = 0\} & ⊂ (S − D)_{(0,κ_{∞})} \\ H_{∞} = \{κ_{∞} = 0\} & : & C_{∞,(κ_{0},0)} := \{y_{0} = y_{2} = 0\} & ⊂ (S − D)_{(κ_{0},0)} \end{aligned} (6.2)

Then now it is easy to see that the Painlevé system (6.1) can be reduced to the following Riccati equations on C_{0} and C_{∞} respectively.

• On C_{0,(0,κ_{∞})} ∩ U_{0}_{(0,κ_{∞})}

x_{0} ≡ 0, \frac{dy_{0}}{dt} = −2y_{0}^{2} + 2ty_{0} − κ_{∞}.  (6.3)

• On C_{∞,(κ_{0},0)} ∩ U_{0}_{(κ_{0},0)}
\frac{dx_{0}}{dt} = −x_{0}^{2} − 2tx_{0} − 2κ_{0}, y_{0} ≡ 0.  (6.4)
fiber  (−2)-curves configuration
(S − D)_{(0,0)} {C_{0}, C_{∞}}  A_{2}

Let us consider the neighborhood of (κ_{0}, κ_{∞}, t) = (0, 0, t) ∈ M_{R} × B_{R} and the hyperplanes as in (6.2). Then, over the subvariety H_{0} ∩ H_{∞} = {(0, 0, t)}, the family (S − D)_{0,0} contains both of families of nodal curves C_{0} ∪ C_{∞} (A_{2}- configuration), (see Figure 3 ). We call this phenomenon the confluence of nodal curves of Okamoto–Painlevé pairs.

Besides hyperplanes H_{0}, H_{∞}, we also have the hyperplane

H_{κ_{0}=κ_{∞}} = {κ_{0} = κ_{∞}}.

Then one can easily see that over hyperplane H_{κ_{0}=κ_{∞}} there exists a family of (−2)-curves defined by

C_{κ_{0}=κ_{∞}} ∩ Ũ_{0}_{κ_{0}=κ_{∞}} = {x_{0}y_{0} − κ_{0} = 0}.

Note that if κ_{0} goes to 0, then the defining equation of the family becomes x_{0}y_{0} = 0. Therefore on (S − D)_{0,0} we have a homological relation:

C_{κ_{0}=κ_{∞}} = C_{0} ∪ C_{∞}.

(See Figure 3). On C_{κ_{0}=κ_{∞}}, the Painlevé system (6.1) can be reduced to

\frac{dx_{0}}{dt} = −x_{0}^{2} − 2tx_{0} + 2κ_{0},  (6.5)
\frac{dy_{0}}{dt} = −2y_{0}^{2} + 2ty_{0} + κ_{0}, .  (6.6)

Note that if κ_{0} ≠ 0 the equations (6.5) and (6.6) can be transformed to each other by the coordinate change x_{0} = κ_{0}/y_{0}.

The hyperplanes are reflection hyperplanes in M_{R} with respect to the reflections of the affine Weyl group W (Ã_{2}), which acts on both M_{R} or S as Bäcklund transformations (cf. [U-W1] and [NTY]). For example, by Bäcklund transformations, the Riccati equations (6.3), (6.4) and (6.5) are birational equivalent to each other. See Theorem 3.3 in [U-W1].

6.2. Rational solutions

We shall remark briefly on rational solutions of Painlevé equations. In the above example, when (κ_{0}, κ_{∞}) = (0, 0), the functions

(x_{0}, y_{0}) ≡ (0, 0)  (6.7)

C_{0} _{C_{∞}}_{κ_{∞}}  _{κ_{0}}_{κ_{∞} = 0}_{κ_{0} = 0}  _{A_{2}}_{A_{1}}  _{A_{1}}  _{A_{1}} κ_{0} − κ_{∞} = 0

Figure 3. A Confluence of Nodal Curves in the case Ẽ_{6} (P_{IV}).

34 Masa-Hiko Saito and Hitomi Terajima

C_{0} C_{∞}  t ∈ B_{R} Y  Y  S

Figure 4. Rational solution coming from C_{0} ∩ C_{∞} for Ẽ_{6} (P_{IV}).

give a solution of the system (6.1), hence gives a rational solution for the Painlevé equation P_{IV}. From the view point of the geometry of Okamoto– Painlevé pairs, it is clear that the intersection of two different families of nodal curves C_{0}, C_{∞} gives a solution of the Painlevé equation. In fact, Painlevé vector field ṽ in (2.4) is tangent to each family of rational curves by Proposition 2.1, hence tangent to their intersection. (See Figure 4). It is not surprising that not all rational solutions of Painlevé equations can be obtained in this way. For example, as we explained after Theorem 5.1, the equation S_{II}(0) in (5.8) has the rational solution (x_{0}, y_{0}) = (0, \frac{t}{2}), but no Riccati solution. It should be an interesting problem to understand the rational or algebraic solutions from the view point of the geometry of Okamoto–Painlevé pairs.

Here we only remark that there are many works for the classification problems of rational and algebraic solutions. (See e.g., [DM], [Maz], [Mu1], [NO], [O3], [U-W1], [U-W2]).

Appendix A. Local cohomology group H_{D}^{1}(Θ_{S}(− log D))

Let (S, Y) be a rational Okamoto–Painlevé pair of non-fibered type and of additive type which corresponds to Painlevé equations (i.e. of type D̃_{i}(4 ≤ i ≤ 8) or Ẽ_{i}(6 ≤ i ≤ 8)), and set D = Y_{red}.

Applying the classification of nodal curves on S − D, we will investigate the local cohomology group H_{D}^{1}(Θ_{S}(− log D)). Note that the local cohomology group can be regarded as the space of time variables for differential equations associated to (S, D) (cf. §3. [STT]).

We state our conjecture for the local cohomology:

Conjecture A.1 (Conjecture 3.1. [STT], [T1]). Let (S, Y) be a rational Okamoto-Painlevé pair (S, Y) as above. Then we have

H_{D}^{1}(Θ_{S}(− log D)) ≃ C.  (A.1)

For the positivity of the dimension of the cohomology group, we have the following result:

Theorem A.1 (Theorem 2.1. [T1]).

dim H^{0}(D, Θ_{S}(− log D) ⊗ N_{D}) = 1.  (A.2)

Here we put N_{D} = O_{S}(D)/O_{S}.

In particular, a natural inclusion

H^{0}(D, Θ_{S}(− log D) ⊗ N_{D}) ↪ H_{D}^{1}(Θ_{S}(− log D)),

implies

dim H_{D}^{1}(Θ_{S}(− log D)) ≥ 1.  (A.3)

On the other hand, in this section, we shall prove

Theorem A.2. Let

S  ↩ D
π ↓  ↙ φ
M_{R} × B_{R}

be the semi-universal deformation of rational Okamoto–Painlevé pairs (S, D) whose type is one of Ẽ_{8}, Ẽ_{7}, D̃_{8}, D̃_{6}, Ẽ_{6}, D̃_{5} and D̃_{4} (i.e., except for R = D̃_{7}). Then there is a Zariski open set U ⊂ M_{R} × B_{R} such that for any (α, t) ∈ U,

dim H_{D_{(α,t)}}^{1}(Θ_{S_{(α,t)}}(− log D_{(α,t)}) = 1.

Remark 8. For (S, Y) of type D̃_{8} or Ẽ_{8}, Theorem A.1 proves Conjecture A.1. In fact, we always have the inclusion

H_{D}^{1}(Θ_{S}(− log D)) ↪ H^{1}(S, Θ_{S}(− log D))

and dim H^{1}(S, Θ_{S}(− log D)) = 10 − 9 = 1 for these cases.

From Remark 8, in order to show Theorem A.2, we will estimate the dimension of the local cohomology group for a special rational Okamoto–Painlevé pairs of other type R.

We first calculate some cohomology groups.

Lemma A.1. Let (S, Y) be a rational Okamoto–Painlevé pair, and C a normal crossing divisor of S. Moreover, let C = ∑_{i=1}^{s} C_{i} be an irreducible decomposition of C, and we assume that {C_{i}}_{i=1}^{s} is linearly independent in H^{2}(S, C) ≃ Pic(S) ⊗ C. Then we have

H^{2}(S, Θ_{S}(− log C)) = {0}.

Proof. We have only to replace D of [Lemma 2.2 and Corollary 2.1,[STT]] with C.

Lemma A.2. Let (S, Y) be a generalized rational Okamoto–Painlevé pair such that D = Y_{red} = ∑_{i=1}^{r} Y_{i} is a normal crossing divisor with at least two irreducible components, say r ≥ 2, and let C = ∑_{i=1}^{s} C_{i} be a normal crossing divisor of S. We assume that

1. C ⊂ S − D,

2. C_{i} ≃ P^{1},
3. {Y_{i}, C_{j}|1 ≤ i ≤ r, 1 ≤ j ≤ s} is linearly independent.

Then we have

dim H^{1}(S, Θ_{S}(− log(D + C))) = 10 − (r + s).

Proof. Note that assumption 1 implies D + C is normal crossing and K_{S} · C_{i} = −Y · C_{i} = 0. We have H^{2}(S, Θ_{S}(− log(D + C))) = 0 by applying Lemma A.1 to D + C. Therefore by using the same argument as Proposition 2.2 in [STT], we have the assertion.

Remark 9. We have the following exact sequence of sheaves:

0 → Θ_{S}(− log(D + C)) → Θ_{S}(− log(D + C − C_{i})) → N_{C_{i}/S} → 0

where N_{C_{i}/S} = O_{S}(C_{i})/O_{S} denotes the normal bundle of the divisor C_{i} ⊂ S. Note that since N_{C_{i}/S} = O_{C_{i}}(−2), we have H^{0}(N_{C_{i}/S}) = {0}. Then the morphism

H^{0}(Θ_{S}(− log(D + C))) → H^{0}(Θ_{S}(− log(D + C − C_{i})))

is injective. Moreover we have

dim H^{0}(Θ_{S}(− log(D + C − C_{i}))) − dim H^{0}(Θ_{S}(− log(D + C))) = 1

by Lemma A.2. This implies that there exist a deformation (S′, D′) of (S, D) such that only the curve C_{i} vanish and other nodal curves remain.

Now we obtain the following

Proposition A.1. Let (S, Y) be a rational Okamoto–Painlevé pair “of non-fibered type ” such that D = Y_{red} is a normal crossing divisor with at least two irreducible components, say r ≥ 2. We suppose the existence of a divisor C = ∑_{i=1}^{9−r} C_{i} of S satisfying the conditions in Lemma A.2. Then we have

dim H_{D}^{1}(Θ_{S}(− log D)) ≤ 1.

Proof. Let us consider the following exact sequence of local cohomology groups (cf. [Corollary 1.9,[Gr]])

H^{0}(S−D, Θ_{S}(− log(D+C))) → H_{D}^{1}(Θ_{S}(− log(D+C))) → H^{1}(S, Θ_{S}(− log(D+C))).

We have an inclusion

H^{0}(S − D, Θ_{S}(− log(D + C))) ↪ H^{0}(S − D, Θ_{S}(− log D)) = H^{0}(S − D, Θ_{S}).

Since (S, Y) is of non-fibered type, from (2) of Proposition 2.1 in [STT], we have H^{0}(S − D, Θ_{S}) = {0}. Therefore we have

H^{0}(S − D, Θ_{S}(− log(D + C))) = {0}.

By applying Lemma A.2, we see

H^{1}(S, Θ_{S}(− log(D + C))) ≃ C.

Moreover, since C ⊂ S − D, we have H_{D}^{1}(Θ_{S}(− log D)) ≃ H_{D}^{1}(Θ_{S}(− log(D + C))), which proves the assertion.

Lemma A.3. For the types D̃_{4}, D̃_{5}, D̃_{6}, Ẽ_{7} and Ẽ_{8}, there exists a rational Okamoto–Painlevé pair (S, Y) of non-fibered type satisfying the assumption of Proposition A.1.

Proof. For each case, we only have to show the existence of nodal curves C_{j} ⊂ S − D j = 1, ⋯, 9 − r on a rational Okamoto–Painlevé pair (S, Y) of non-fibered type. The existence of (−2)-curves follows from Theorem 3.1.

Remark 10. For any rational Okamoto–Painlevé pair (S, Y) of D̃_{7}, there is no (−2)-curve C on S − D satisfying the condition in Lemma A.2 (cf. Table 4, 2).

Lemma A.3 and Theorem A.1 lead us the following corollary, which also implies Theorem A.2.

Corollary A.1. For the types D̃_{4}, D̃_{5}, D̃_{6}, D̃_{8}, Ẽ_{7} and Ẽ_{8}, there exists a rational Okamoto–Painlevé pair (S, Y) of non-fibered type such that

dim H_{D}^{1}(Θ_{S}(− log D)) = 1.

Acknowledgements

We would like to thank Tetsu Masuda, Masatoshi Noumi, Kyoichi Takano, Yasuhiko Yamada and Kota Yoshioka for valuable discussions during the preparation of this paper.

Department of Mathematics

Faculty of Science, Kobe University

Rokko, Kobe, 657-8501

Japan

Research Institute for Mathematical Sciences Kyoto University Kyoto, 606-8502 JAPAN

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