CLASSIFICATION OF OKAMOTO–PAINLEVÉ PAIRS

MASA-HIKO SAITO AND TARO TAKEBE

Abstract. In this paper, we introduce the notion of an Okamoto–Painlevé pair (S, Y) which consists of a compact smooth complex surface S and an effective divisor Y on S satisfying certain conditions. Though spaces of initial values of Painlevé equations introduced by K. Okamoto gives examples of Okamoto–Painleve pairs, we find a new example of Okamoto–Painlevé pairs not listed in [Oka]. We will give the complete classification of Okamoto–Painlevé pairs.

0. Introduction

In this paper, we will introduce the notion of an Okamoto–Painlevé pair (S, Y), which is defined as follows: Definition 0.1. (Cf. Definition 2.1). Let S be a compact smooth complex surface and Y = ∑_{i=1}^{r} a_{i}Y_{i} an effective divisor on S. We say that a pair (S, Y) is an Okamoto–Painlevé pair if it satisfies the following conditions:

(i) There exists a meromorphic 2-form ω on S such that (ω) = −Y, that is, ω has the pole divisor Y (counting multiplicities) and has no zero outside Y.

(ii) For all i (1 ≤ i ≤ r), Y · Y_{i} = deg[Y]|_{Y_{i}} = 0.

(iii) Let us set D := Y_{red} = ∑_{i=1}^{r} Y_{i}. Then S − D contains C^{2} as a Zariski open set.

(iv) Set F = S − C^{2} where C^{2} is the same Zariski open set as in (ii). Then F is a (reduced) divisor with normal crossings.

Historically, Okamoto [Oka] introduced the space M_{J}(t) of initial values for each Painlevé equation of type P_{J} (J = I, …, V I) with the time parameter t, which is a noncompact complex surface. We can obtain a nice compactification \overline{M}_{J}(t) of M_{J}(t) so that (\overline{M}_{J}(t), D_{J}(t)) becomes an Okamoto–Painlevé pair, where D_{J}(t) = \overline{M}_{J}(t)−M_{J}(t) (cf. [ST] and [MMT]). Conversely, for each Okamoto–Painlevé pair one can associate a Hamiltonian system via the deformation theory of pairs, and such a Hamiltonian system is equivalent to a differential equation of Painlevé type (cf. [Sa–T]).

The main purpose of this paper is the classification of Okamoto–Painlevé pairs. We will classify Okamoto–Painlevé pairs (S, Y) into seven types according to the configuration of the divisor Y ∈ | − K_{S}|. (See Theorem 2.1). One can easily see that the configuration of the divisor Y is same as one of singular fibers of elliptic surfaces [Kod1]. According to the Kodaira’s classification, the seven configurations can be denoted by II^{∗}, III^{∗}, IV^{∗}, I_{3}^{∗}, I_{2}^{∗}, I_{1}^{∗}, I_{0}^{∗}, or in the notation of the dual graph of Y they can be denoted by

Ẽ_{8}, Ẽ_{7}, Ẽ_{6}, D̃_{7}, D̃_{6}, D̃_{5}, D̃_{4},

respectively.

It is known that there is a one-to-one correspondence between Okamoto–Painlevé pairs of the six types Ẽ_{8}, Ẽ_{7}, Ẽ_{6}, D̃_{7}, D̃_{6}, D̃_{5}, D̃_{4} and Painlevé equations of six types P_{I}, P_{II}, P_{III}, P_{IV}, P_{V}, P_{V I}, respectively (cf. [Oka], [MMT] and [ST]). Though an Okamoto–Painlevé pair of type D̃_{7} did not appear in the list of Okamoto [Oka], we can obtain a Hamiltonian system associated to the pairs. Recently, Okamoto informed us that he found Painlevé equations for an Okamoto–Painlevé pair of type D̃_{7} which are special cases of P_{III}. Therefore, we denote this Painlevé equation by P_{III}^{∗}. In §4, we will construct an Okamoto–Painlevé pair of type D̃_{7} by blowing-up of F_{0} = P^{1} × P^{1} with centers on the anti-canonical divisor −K_{F_{0}}.

Partly supported by Grant-in Aid for Scientific Research (B-09440015), the Ministry of Education, Science and Culture, Japan.

Let us discuss main ideas of classfying Okamoto–Painlevé pairs. For an Okamoto–Painlevé pair (S, Y), the surface S is the compactification of C^{2}, namely S includes C^{2} as a Zariski open set. Let F = S − C^{2}. By the definition of Okamoto–Painlevé pairs, F is a normal crossing divisor. Each component of Y_{red} is also a component of F, namely,

Y_{red} = ∑_{i=1}^{r} Y_{i} ⊂ ∑_{i=1}^{l} F_{i} = F,

hence Y_{red} is also a normal crossing divisor. One can easily see that the divisor Y has the same configuration of a singular fiber of an elliptic surface, and the number of irreducible components of F is 10. Therefore, the number of the irreducible components of Y_{red} is less than or equal to 10. Moreover, the configuration of F must be a tree, so is Y_{red}. These arguments show that the configuration of Y is one of the types Ẽ_{r−1} for r = 9, 8, 7 and D̃_{r−1} for r = 5, …, 10.

Now we will consider the classification of the configurations of normal clossing divisors F. The divisor F can be obtained by adding some components to Y_{red}. We call an irreducible component of F − Y_{red} an additional component. We will classify all of configurations of F, and this gives the complete classification of Okamoto–Painlevé pairs and also the configurations of F.

The complete list of all normal crossing divisors F is given in §5. Each of those is a divisor with ten components including one of the seven Kodaira types. All configurations of F in the list can be transformed into the anti-canonical divisors of P^{2}, F_{0} and F_{2} by blowings-up and blowings-down.

Let us discuss the relation between our results and results of Sakai in [Sak]. Sakai defined the notion of generalized Halphen surfaces. By definition, a generalized Halphen surface S is a compact complex surface satisfying the condition of (i) and (ii) of Definition 0.1. He related generalized Halphen surfaces to the discrete Painlevé equations via Cremona transformations. The Painlevé differential equations can be obtained as limits of discrete Painlevé equations. Most essential extra conditions of an Okamoto–Painlevé pair (S, Y) are that S is a compactification of C^{2} and F is a normal crossing divisor. Our classification gives more direct correspondences between Okamoto–Painlevé pairs and Painlevé differential equations (cf. [Sa–T]).

1. Preliminary

Let S be a compact complex surface and let −K_{S} denote the anti-canonical divisor class of S. For any divisor D on S, we denote by [D] linear equivarence class of D, and by |D| = |[D]| the linear space of effective divisors C, such that C ∼ D. Here “∼” means the linear equivalence of divisors. In this paper, we often identify the divisor class [D] with the isomorphism class of the corresponding line bundle [D] or the correponding invertible sheaf O_{S}(D).

Let us assume that there exists a normal crossing effective divisor Y ∈ | − K_{S}|. Moreover, assume that every irreducible component of Y is a smooth rational curve. Set Y = ∑_{i=1}^{n} m_{i}C_{i}. Consider the blowing-up π : S̃ → S with center of P ∈ C_{i_{0}}, and let C′_{i} be the strict transform of C_{i} by π, and E the exceptional curve of π. Note that m_{i} ≥ 0. By a standard calculation, we can obtain the following lemma.

Lemma 1.1. If p ∈ C_{i_{0}} \ ⋃_{i≠i_{0}}(C_{i_{0}} ∩ C_{i}), then

−K_{S} = [∑_{i=1}^{n} m_{i}C′_{i} + (m_{i_{0}} − 1)E],

(C′_{i})^{2} = (C_{i})^{2} (i ≠ i_{0}),

and

(C′_{i_{0}})^{2} = (C_{i_{0}})^{2} − 1.

If p ∈ C_{i_{0}} ∩ C_{i_{1}} for some i_{1} (≠ i_{0}), then

−K_{S} = [∑_{i=1}^{n} m_{i}C′_{i} + (m_{i_{0}} + m_{i_{1}} − 1)E],

(C′_{i})^{2} = (C_{i})^{2} (i ≠ i_{0}, i_{1}),

and

(C′_{i})^{2} = (C_{i})^{2} − 1 (i = i_{0}, i_{1}).

Let F_{n} denote the Hirzebruch surface P(O_{P^{1}} ⊕ O_{P^{1}}(n)). We denote by f the class of fiber of F_{n}, s_{0} the class of the minimal section of F_{n} respectively. We also set s_{∞} = s_{0} + nf. Then, the self-intersection numbers are given by

f^{2} = 0, (s_{0})^{2} = −n, (s_{∞})^{2} = n.

It is well-known that F_{n} can be obtained by performing several blowings-up and blowing-downs to P^{2}.

Now we consider the anti-canonical divisor class of P^{2} and F_{n}. Let h denote the class of lines in P^{2}. Then, they are given as

−K_{P^{2}} = 3h,
−K_{F_{n}} = 2s_{0} + (n + 2)f = 2s_{∞} − (n − 2)f.

In this paper, we often use the formula −K_{F_{0}} = 2s_{0} + 2f and −K_{F_{2}} = 2s_{∞}. Note that −K_{F_{0}} and −K_{F_{2}} have complements in F_{0} and F_{2} respectively which contain C^{2} as Zariski open sets.

2. Okamoto–Painlevé Pairs

Now let us give the definition of Okamoto–Painlevé pairs. Definition 2.1. Let S be a compact smooth complex surface and Y = ∑_{i=1}^{r} a_{i}Y_{i} an effective divisor on S. We say that a pair (S, Y) is an Okamoto–Painlevé pair if it satisfies the following conditions:

(i) There exists a meromorphic 2-form ω on S such that (ω) = −Y, that is, ω has the pole divisor Y (counting multiplicities) and has no zero outside Y.

(ii) For all i (1 ≤ i ≤ r), Y · Y_{i} = deg[Y]|_{Y_{i}} = 0.

(iii) Let us set D := Y_{red} = ∑_{i=1}^{r} Y_{i}. Then S − D contains C^{2} as a Zariski open set.

(iv) Set F = S − C^{2} where C^{2} is the same Zariski open set as in (ii). Then F is a (reduced) divisor with normal crossings. Remark 2.1. (i) The meromorphic 2-form ω as above induces a holomorphic symplectic structure on S − D.

(ii) Let K_{S} denote the canonical divisor class of S. The condition (i) means that K_{S} = [−Y] or −K_{S} = [Y].

(iii) The condition (v) implies that the reduced part D = Y_{red} of Y is also a divisor with normal crossings.

(iv) The spaces of initial values of Painlevé equations can be written as S − D for some Okamoto– Painlevé pair (S, Y). (See [Oka], [ST] and [MMT]).

The following is the main theorem of this paper. Theorem 2.1. Let (S, Y) be an Okamoto–Painlevé pair. Then we have the following assertions.

(i) The surface S is a projective rational surface.

(ii) The configuration of Y counting multiplicities is in the list of Kodaira’s classification of singular fibers of elliptic surfaces. More precisely, they are given by one of the following lists.

(iii) All Okamoto–Painlevé pairs can be obtained by blowings-up and blowings-down of (P^{2}, 3H), where H is a line in P^{2}.

Y  Ẽ_{8}  Ẽ_{7}  D̃_{7}  D̃_{6}  Ẽ_{6} D̃_{5} D̃_{4}

Kodaira’s notation II^{∗} III^{∗} I_{3}^{∗} I_{2}^{∗} IV^{∗} I_{1}^{∗} I_{0}^{∗}

Painlevé equation P_{I} P_{II} P_{III}^{∗} P_{III} P_{IV} P_{V} P_{V I}

1  2 3  4 5  6 3 4  2 Ẽ_{8}  1  2 3 4 2  3  2 1 Ẽ_{7}

3 2 1 2 1  2  1 Ẽ_{6}

2 1  1 2  2 2  1 1 D̃_{7}  2 2 1 1  2  1 1 D̃_{6}

2 1  1  2  1 1 D̃_{5}  2 1  1  1  1 D̃_{4} Figure 1.

In Figure 1, each line denotes P^{1} whose self-intersection number is equal to −2, and the number next to each line denotes the multiplicity of corresponding component in −K_{S}.

Let us start our proof with the following easy lemmas.

Lemma 2.1. Let (S, Y) be an Okamoto–Painlevé pair, then we have:

(i) Y_{i} ≅ P^{1} for i (1 ≤ i ≤ r). (ii) The following conditions are equivarent for i (1 ≤ i ≤ r).

(a) Y · Y_{i} = 0.

(b) (Y_{i})^{2} = −2.

Proof. (i) It is sufficient to show that the component F_{i} of F is isomorphic to P^{1}. We will show that H^{1}(F_{i}, Z) = 0.

We have the following isomorphisms of cohomology groups with Z-coefficients by Poincaré duality.

H^{i}(S, F; Z) ≅ H_{c}^{i}(C^{2}, Z)
≅ H_{4−i}(C^{2}, Z)
≅ \{ \begin{matrix}0 & i = 0, 1, 2, 3 \\ Z & i = 4 \end{matrix}

On the other hand, consider the long exact sequence of cohomology groups for pair (S, F)

0 → H^{0}(F, Z) → H^{0}(S, Z) → H^{0}(S, F; Z) = 0
→ H^{1}(F, Z) → H^{1}(S, Z) → H^{1}(S, F; Z) = 0
→ H^{2}(F, Z) → H^{2}(S, Z) → H^{2}(S, F; Z) = 0
→ H^{3}(F, Z) → H^{3}(S, Z) → H^{3}(S, F; Z) = 0
→ H^{4}(F, Z) → H^{4}(S, Z) → H^{4}(S, F; Z) ≅ Z
→ ….

Then we have

H^{i}(S, Z) ≅ H^{i}(F, Z) (i = 0, 1, 2, 3).  (1)

Especially,

H^{3}(S, Z) ≅ H^{3}(F, Z) = 0.

By Poincaré duality again, we have

H^{1}(S, Z) ≅ H^{3}(S, Z) = 0.  (2)

Now we consider an irreducible component F_{i} of F. Let F′ = ⋃_{k≠i} F_{k}. Then F_{i} ∩ F′ consists of a finite set of points. And we have the Mayer–Vietoris exact sequence

0 → H^{0}(F, Z) → H^{0}(F_{i}, Z) ⊕ H^{0}(F′, Z) → H^{0}(F_{i} ∩ F′, Z)
→ H^{1}(F, Z) → H^{1}(F_{i}, Z) ⊕ H^{1}(F′, Z) → H^{1}(F_{i} ∩ F′, Z)
→ ….

(3)

By (1) and (2), we have H^{1}(F, Z) = 0. Since H^{1}(F_{i} ∩ F′, Z) = 0, we have H^{1}(F_{i}, Z) ⊕ H^{1}(F′, Z) = 0. Therefore, H^{1}(F_{i}, Z) = 0. If the irreducible component F_{i} is a singular nodal curve, then H^{1}(F_{i}, Z) ≠ 0. So F_{i} is nonsingular, namely F_{i} ≅ P^{1}.

(ii) Since Y_{i} is a nonsingular rational curve, by the adjunction formula, we have K_{S} ·Y_{i}+(Y_{i})^{2} = −2. Since K_{S} = [−Y], the conditions (a) and (b) are equivalent to each other. Lemma 2.2.

H^{2}(F, Z) ≅ ⨁_{i=1}^{l} H^{2}(F_{i}, Z) ≅ ⨁_{i=1}^{l} Z[F_{i}].

Proof. Let F′ = ⋃_{k≠1} F_{k}. We see that H^{1}(F_{1} ∩F′, Z) = H^{2}(F_{1} ∩F′, Z) = 0. From the exact sequence (3), we have

H^{2}(F, Z) ≅ H^{2}(F_{1}, Z) ⊕ H^{2}(F′, Z).

Using the argument inductively, we obtain the assertion. Lemma 2.3. The configuration of F is a tree, that is, the dual graph of F is connected and contains no cycles. Proof. Since H^{0}(S, Z) ≅ H^{0}(F, Z), the connectivity of S implies the connectivity of F. To show that F contains no cycles, it is sufficient to show that there are no irreducible components F_{i_{1}}, …, F_{i_{m}} (m ≥ 3) such that

\begin{matrix}F_{i_{k}} ∩ F_{i_{k+1}} ≠ ∅ & (k = 1, …, m − 1), \\ F_{i_{m}} ∩ F_{i_{1}} ≠ ∅. \end{matrix}

We assume that there exist such F_{i_{1}}, …, F_{i_{m}}. Let F^{(1)} = F_{i_{1}} ∪ ⋯ ∪ F_{i_{m}} and F^{(2)} = F_{i_{1}} ∪ ⋯ ∪ F_{i_{m−1}}. Since F^{(1)} and F^{(2)} are connected, we have H^{0}(F^{(1)}, Z) ≅ H^{0}(F^{(2)}, Z) ≅ Z. F^{(2)} ∩ F_{i_{m}} consists of at least two points, and let ν be the number of the points of F^{(2)}∩F_{i_{m}}. Then we have H^{0}(F^{(2)}∩F_{i_{m}}, Z) ≅ Z^{ν} (ν ≥ 2). Note that H^{0}(F_{i_{m}}, Z) ≅ H^{0}(P^{1}, Z) ≅ Z. We consider the Mayer–Vietoris exact sequence

0 → \underbrace{H^{0}(F^{(1)}, Z)}_{≅Z} → \underbrace{H^{0}(F^{(2)}, Z) ⊕ H^{0}(F_{i_{m}}, Z)}_{≅Z⊕Z} → \underbrace{H^{0}(F^{(2)} ∩ F_{i_{m}}, Z)}_{≅Z^{ν}}
→ H^{1}(F^{(1)}, Z) → H^{1}(F^{(2)}, Z) ⊕ H^{1}(F_{i_{m}}, Z) → ….

If we assume H^{1}(F^{(1)}, Z) = 0 then ν must be one, which contradicts the fact that ν ≥ 2. So we have H^{1}(F^{(1)}, Z) ≠ 0. Now let F^{(3)} be the union of irreducible components of F which do not belong to F^{(1)}. Namely, F = F^{(1)} ∪ F^{(3)}. We consider the exact sequence

\underbrace{H^{1}(F, Z)}_{=0} → H^{1}(F^{(1)}, Z) ⊕ H^{1}(F^{(3)}, Z) → H^{1}(F^{(1)} ∩ F^{(3)}, Z).

Since F^{(1)} ∩ F^{(3)} consists of a finite set of points, we have H^{1}(F^{(1)} ∩ F^{(3)}, Z) = 0, so H^{1}(F^{(1)}, Z) ⊕ H^{1}(F^{(3)}, Z) = 0. Therefore we have H^{1}(F^{(1)}, Z) = 0, which is a contradiction. This completes the proof.

Proof of Theorem 2.1. (i) Since S is a compactification of C^{2}, we see that S is a rational surface by Theorem 5 of Kodaira [Kod2].

(ii) Let (S, Y) be an Okamoto–Painlevé pair. We know that S is a projective rational surface and the configuration of the dual graph of F = S − C^{2} is a tree. Let F = ∑_{i=1}^{l} F_{i} be the irreducible decomposition of F, where l denotes the number of the irreducible components of F. Let [F_{i}] denote the class in H^{2}(S, Z) dual to the class of F_{i} ⊂ S. From Lemma 2.2 and (1), we obtain the isomorphism

⨁_{i=1}^{l} Z[F_{i}] ≅ H^{2}(F, Z) →^{≅} H^{2}(S, Z).

This implies that {[F_{i}] | 1 ≤ i ≤ l} forms an integral basis of H^{i}(S, Z).

From the condition (iii) of Definition 0.1, we have

Y^{2} = ∑_{i=1}^{r} a_{i}(Y · Y_{i}) = 0.

Since K_{S} = −Y, we know that (K_{S})^{2} = (−Y)^{2} = Y^{2} = 0. By Noether’s formula, we have

e(S) + (K_{S})^{2} = 12χ(S, O_{S}),

and χ(S, O_{S}) = 1. Then we obtain

e(S) = 2 − 2b_{1}(S) + b_{2}(S) = 12.

Since b_{1}(S) = 0, we have b_{2}(S) = 10, and hence

l = (the number of the components of F) = 10.

Since D = ∑_{i=1}^{r} Y_{i} ⊂ F = ∑_{i=1}^{10} F_{i}, we have

r ≤ 10.

Since F = ∑_{i=1}^{10} F_{i} is a connected divisor with normal crossings, the divisor D = Y_{red} = ∑_{i=1}^{r} Y_{i} is also a connected divisor with normal crossings. To see that D = Y_{red} is connected, it is sufficient to show that Y is connected. By Serre duality, we have H^{1}(S, O_{S}) ≅ H^{1}(S, O(K_{S})). Hence we have H^{1}(S, O(K_{S})) = 0. By the exact sequence

0 → O(K_{S}) → O_{S} → O_{Y} → 0,

we obtain the long exact sequence

\begin{aligned}0 & → & \underbrace{H^{0}(S, O(K_{S}))}_{=0} & → & H^{0}(S, O_{S}) & → & H^{0}(S, O_{Y}) \\ & → & \underbrace{H^{1}(S, O(K_{S}))}_{=0} & → & H^{1}(S, O_{S}) & → & …. \end{aligned}

Therefore we have H^{0}(S, O_{Y}) ≅ H^{0}(S, O_{S}) ≅ C. This shows that Y is connected.

Now we prove that the divisor Y = ∑_{i=1}^{r} a_{i}Y_{i} has one of the configurations in the list of singular fibers of elliptic surfaces (cf. Theorem 6.2, [Kod1]). First let us show that the greatest common divisor of {a_{i}}_{i=1}^{r} is equal to one. Since S is a rational surface with b_{2}(S) = 10, S is not relatively minimal. Hence S contains an exceptional curve E of the first kind, that is, E ≅ P^{1} and E^{2} = −1. Note that K_{S} = [−Y]. By the adjunction formula, we have K_{S} · E + E^{2} = −2 and this implies that K_{S} · E = −1 and equivalently Y · E = 1. Since K_{S} = [−Y] and Y · Y_{i} = 0 for every irreducible component Y_{i} of Y, we know that E is not a component of Y hence E · Y_{i} ≥ 0. On the other hand, since we have

1 = Y · E = ∑_{i=1}^{r} a_{i}(Y_{i} · E),

there exists an irreducible component Y_{i} with a_{i} = 1. Under the condition, in order to see that the configuration of Y is one in the list of the singular fibers of elliptic surfaces in [Kod1], we can follow the proof of Theorem 6.2 of [Kod1]. In fact, only the following conditions are needed to determine all of the configuration of singular fibers of elliptic surfaces:

(1) D = Y_{red} is connected.

(2) K_{S} · Y = 0.
(3) Y · Y_{i} = 0, for i (1 ≤ i ≤ r) .

We have proved the first assertion, and the third assertion follows from Definition 0.1. The second assertion follows from the third assertion because K_{S} = [−Y]. Hence the configuration of Y is in the list of Kodaira. Next, we note that D = Y_{red} must be a divisor with normal crossings and the dual graph of D must be a tree. Then Y must be one of the types of Ẽ_{r−1} for r = 9, 8, 7 and D̃_{r−1} for r = 5, …, 10. At this moment, we see that each irreducible component Y_{i} of D = Y_{red} is a smooth rational curve with (Y_{i})^{2} = −2.

We will show that the configurations of types D̃_{9} and D̃_{8} can not occur. Let us set Λ_{Y} = ∑_{i=1}^{r} Z[Y_{i}] and Λ_{F} = ∑_{j=1}^{10} Z[F_{j}] ≅ Pic(S) ≅ H^{2}(S, Z). Then we have the inclusion map

ι : Λ_{Y} ↪ Λ_{F} ≅ H^{2}(S, Z).

By this inclusion map ι, Λ_{Y} can be considered as a sublattice of H^{2}(S, Z) and H^{2}(S, Z) is generated by [F_{j}] (1 ≤ j ≤ 10). The intersection matrix I_{F} := ((F_{i} · F_{j}))_{1≤i,j≤10} is the unimodular matrix with the signature (b_{+}, b_{−}) = (1, 9). On the other hand, the intersection matrix

I_{Y} = ((Y_{i} · Y_{j}))_{1≤i,j≤r}

has a null eigenvalue corresponding to Y because Y^{2} = 0. This means that the rank of Λ_{Y} is strictly less than ten, that is,

r < 10.

This proves that the configuration of type D̃_{9} does not occur. If Y is of type D̃_{8}, then r = 9 and hence we can write F = ∑_{i=1}^{9} Y_{i} + F_{10}. Since F is connected and the dual graph of F is a tree, we see that F_{10} intersects only one irreducible component Y_{i} for some i (1 ≤ i ≤ 9). In this case, Λ_{F} = H^{2}(S, Z) is generated by Y_{1}, …, Y_{9} and F_{10} where the dual graph of Y_{1}, …, Y_{9} is of type D̃_{8}. Now by a direct calculation we see that the intersection matrix of {Y_{1}, …, Y_{9}, F_{10}} can not have the determinant −1. Hence D̃_{8} does not occur. Then we see that the dual graph of Y must be one of the following types:

Ẽ_{8}, Ẽ_{7}, Ẽ_{6}, D̃_{7}, D̃_{6}, D̃_{5}, D̃_{4}.

Conversely, if Y is one of the types as above, we can construct Okamoto–Painlevé pair (S, Y) by blowing up and blowing down of (P^{2}, Y = 3H) where H denotes a line in P^{2}. For detail, see §5. This proves our theorem. Remark 2.2. We will classify not only the configurations of Y but also F of an Okamoto–Painlevé pair (S, Y). By using the result, one can show that S − D is covered by a finite number of Zariski open sets {U_{i}} each of which is isomorphic to C^{2} if Y is not of type Ẽ_{8}. (See [ST] and [MMT]).

Example 2.1. We consider the case where the configuration of Y is of type D̃_{4}. Namely, the corresponding Painlevé equation is type P_{V I}. For more details, see [ST]. At first, we take the Hirzebruch surface Σ_{(ε)}^{(2)} which is obtained by gluing four copies of C^{2} via following identification. Note that

Σ_{(0)}^{(2)} = F_{2}.

U_{i} = Spec C[x_{i}, y_{i}] ≅ C^{2} (i = 0, 1, 2, 3)
x_{0} = x_{1},  y_{0} = 1/y_{1},
x_{0} = 1/x_{2}, y_{0} = x_{2}(ε − x_{2}y_{2}), (4)
x_{2} = x_{3},  y_{2} = 1/y_{3}

We consider a fiber space (Σ_{(ε)}^{(2)} × B_{V I}, π, B_{V I}), where B_{V I} = C \ {0, 1}. Let us take

ε = (κ_{0} + κ_{1} + κ_{t} − 1 + κ_{∞})/2,

where κ_{ν} (ν = 0, 1, t, ∞) are complex constants in the Hamiltonian function H_{V I} (cf. [ST]). For any parameter t ∈ B_{V I}, we define a divisor D^{(0)}(t) on Σ_{(ε)}^{(2)} × t:

D^{(0)}(t) = {(x_{1}, y_{1}, t) ∈ U_{1} × t | y_{1} = 0} ∪ {(x_{3}, y_{3}, t) ∈ U_{3} × t | y_{3} = 0}.

Note that (D^{(0)}(t))^{2} = 2, and 2D^{(0)}(t) ∈ | − K_{Σ_{(ε)}^{(2)}}|. And we take four points a_{ν}^{(0)}(t) ∈ D^{(0)}(t) (ν =

0, 1, t, ∞):

a_{ν}^{(0)}(t) = {(x_{1}, y_{1}, t) = (ν, 0, t)} ∈ U_{1} ∩ D^{(0)}(t) (ν = 0, 1, t),
a_{∞}^{(0)}(t) = {(x_{3}, y_{3}, t) = (0, 0, t)} ∈ U_{3} ∩ D^{(0)}(t).

[figure: a_{0}^{(0)}(t); a_{1}^{(0)}(t); a_{t}^{(0)}(t); a_{∞}^{(0)}(t); D^{(0)}(t); x_{0}; y_{0}; x_{1}; y_{1}; x_{2}; y_{2}; x_{3}; y_{3}; Figure 2.]

We perform blowings-up to Σ_{(ε)}^{(2)} × t at a_{ν}^{(0)}(t) for all t ∈ B_{V I}, and let D_{ν}^{(1)}(t) be the exceptional

curves of the blowings-up at a_{ν}^{(0)}(t) for ν = 0, 1, t, ∞. We can take four coordinate systems (z_{ν}, w_{ν}) around the points at infinity of the exceptional curves D_{ν}^{(1)}(t) (ν = 0, 1, t, ∞), where

(z_{ν}, w_{ν}) = ((x_{1} − ν)y_{1}^{−1}, y_{1}) (ν = 0, 1, t),
(z_{∞}, w_{∞}) = (x_{3}y_{3}^{−1}, y_{3}).

Note that we have w_{ν} = 0 on D_{ν}^{(1)}(t) for ν = 0, 1, t, ∞. In order to perform the second blowings-up, let us take four points a_{ν}^{(1)}(t) for ν = 0, 1, t, ∞.

a_{ν}^{(1)}(t) = {(z_{ν}, w_{ν}, t) = (κ_{ν}, 0, t)} ∈ D_{ν}^{(1)}(t) (ν = 0, 1, t, ∞),

[figure: x_{0}; y_{0}; x_{2}; y_{2}; z_{0}; w_{0}; z_{1}; w_{1}; z_{t}; w_{t}; z_{∞}; w_{∞}; a_{0}^{(1)}(t); a_{1}^{(1)}(t); a_{t}^{(1)}(t); a_{∞}^{(1)}(t); D^{(0)}(t); D_{0}^{(1)}(t); D_{1}^{(1)}(t); D_{t}^{(1)}(t); D_{∞}^{(1)}(t); Figure 3.]

Let us perform blowings-up at a_{ν}^{(1)}(t), and denote D_{ν}^{(2)}(t) for the exceptional curves, respectively. We take four coordinate systems (Z_{ν}, W_{ν}) around the points at infinity of D_{ν}^{(2)}(t) for ν = 0, 1, t, ∞, where

(Z_{ν}, W_{ν}) = ((x_{1}y_{1}^{−1} − κ_{0})y_{1}^{−1}, y_{1}), ν = 0, 1, t,
(Z_{∞}, W_{∞}) = ((x_{3}y_{3}^{−1} − κ_{∞})y_{3}^{−1}, y_{3}).

[figure: x_{0}; y_{0}; x_{2}; y_{2}; W_{0}; Z_{0}; W_{1}; Z_{1}; W_{t}; Z_{t}; W_{∞}; Z_{∞}; D^{(0)}(t); D_{0}^{(1)}(t); D_{1}^{(1)}(t); D_{t}^{(1)}(t); D_{∞}^{(1)}(t); D_{0}^{(2)}(t); D_{1}^{(2)}(t); D_{t}^{(2)}(t); D_{∞}^{(2)}(t); Figure 4.]

For the strict transform of D_{ν}^{(i)}(t) by the blowing-up, we also denote by D_{ν}^{(i)}(t), respectively. Let S(t) → Σ_{(ε)}^{(2)} × t be the composition of above eight blowings-up for the parameter t. Then, we see that the configuration of the divisor

D(t) := 2D^{(0)}(t) + ∑_{ν=0,1,t,∞} D_{ν}^{(1)}(t)

on S(t) is of type D̃_{4}. And we see that the complements of D(t) in S(t) is covered by six Zariski open sets

Spec C[Z_{ν}, W_{ν}] (ν = 0, 1, t, ∞),
Spec C[x_{0}, y_{0}],
Spec C[x_{2}, y_{2}].

Note that this example corresponds to the Okamoto–Painlevé pair of type D̃_{4}–(2) in our classification. (See §5).

Example 2.2. We will construct the space of initial values of the Painlevé equation of type P_{I} for an example of the Okamoto–Painlevé pair of type Ẽ_{8}, As we remarked in Remark 2.2, the spaces of initial values of P_{I} can not be covered by some Zariski open sets each of which is isomorphic to C^{2}. For any t ∈ B_{I}, we consider the Hirzebruch surface Σ(t) := Σ_{(0)}^{(2)} = F_{2}, and take two curves D_{0}(t) = {y_{0} = 0} ∪ {y_{2} = 0} and D′_{0}(t) = {x_{2} = 0} ∪ {x_{3} = 0} on Σ(t). Furthermore, we have to consider the multiplicities of each component of the anti-canonical divisor. For the surface Σ(t), we can take the anti-canonical divisor 2D_{0}(t) + 4D′_{0}(t) ∈ | − K_{Σ(t)}|. Let us perform blowing-up at the point a_{0}(t) = {(x_{3}, y_{3}) = (0, 0)}.

[figure: a_{0}(t); D_{0}(t); D′_{0}(t); x_{0}; y_{0}; x_{1}; y_{1}; x_{2}; y_{2}; x_{3}; y_{3}; 4; 2; Figure 5.]

Let D_{1} denote the exceptional curve, and take two coordinate systems (z_{1}, W_{1}) and (Z_{1}, w_{1}) on D_{1} which satisfies D_{1} = {z_{1} = 0} ∪ {w_{1} = 0} and Z_{1} = W_{1}^{−1}. Note that

(z_{1}, W_{1}) = (x_{3}, x_{3}^{−1}y_{3}),
(Z_{1}, w_{1}) = (x_{3}y_{3}^{−1}, y_{3}).

[figure: D_{0}(t); D′_{0}(t); D_{1}(t); z_{1}; W_{1}; Z_{1}; w_{1}; 2; 4; 3; Figure 6.]

Next, let us perform the blowing-up at the point a_{1}(t) = {(Z_{1}, w_{1}) = (0, 0)}, and denote the exceptional curve by D_{2}(t). Take two coordinate systems (z_{2}, W_{2}) and (Z_{2}, w_{2}) on D_{2} such that D_{2} = {z_{2} = 0} ∪ {w_{2} = 0} and Z_{2} = W_{2}^{−1}. Note that

(z_{2}, W_{2}) = (x_{3}y_{3}^{−1}, x_{3}^{−1}y_{3}^{2}),
(Z_{2}, w_{2}) = (x_{3}y_{3}^{−2}, y_{3}).

[figure: Q; P; D_{0}(t); D′_{0}(t); D_{1}(t); D_{2}(t); z_{2}; W_{2}; Z_{2}; w_{2}; 2; 4; 3; 6; Figure 7.]

Let P = P (t) = {(Z_{2}, w_{2}) = (0, 0)} and Q = Q(t) = {(z_{2}, W_{2}) = (0, 0)}. Let us perform blowing-up at the point a_{2}(t) = {(Z_{2}, w_{2}) = (1/4, 0)} = {(z_{2}, W_{2}) = (0, 4)}, and denote the exceptional curve by D_{3}(t). Note that we can consider this blowing-up by using the coordinate system either (Z_{2}, w_{2}) or (z_{2}, W_{2}).

At first, we consider the blowing-up at P with the coordinate system (Z_{2}, w_{2}). Take two coordinate systems (z_{3}, W_{3}) and (Z_{3}, w_{3}) on D_{3}(t) such that D_{3}(t) = {z_{3} = 0} ∪ {w_{3} = 0} and Z_{3} = W_{3}^{−1}. Then we have

\{ \begin{matrix}z_{3} & = & x_{3}y_{3}^{−2} − 1/4, \\ W_{3} & = & (x_{3}y_{3}^{−2} − 1/4)^{−1}y_{3}, \end{matrix}

and

\{ \begin{matrix}Z_{3} & = & (x_{3}y_{3}^{−2} − 1/4)y_{3}^{−1}, \\ w_{3} & = & y_{3}. \end{matrix}

[figure: D_{0}(t); D′_{0}(t); D_{1}(t); D_{2}(t); D_{3}(t); z_{3}; W_{3}; Z_{3}; w_{3}; 2; 4; 3; 6; 5; Figure 8.]

In the same way, we have to perform five more blowings-up, and take the coordinate systems (z_{i}, W_{i}) and (Z_{i}, w_{i}) which satisfies that D_{i}(t) = {z_{i} = 0} ∪ {w_{i} = 0} and Z_{i} = W_{i}^{−1}, where D_{i}(t) is the exceptional curve of each blowing-up (i = 4, 5, 6, 7). Similarly, let a_{i}(t) be the center of each blowingup. Then we have to take a_{i}(t) as follows. For more details, see [Oka].

a_{3}(t) = {(Z_{3}, w_{3}) = (0, 0)},
a_{4}(t) = {(Z_{4}, w_{4}) = (0, 0)},
a_{5}(t) = {(Z_{5}, w_{5}) = (0, 0)},
a_{6}(t) = {(Z_{6}, w_{6}) = (t/2, 0)},
a_{7}(t) = {(Z_{7}, w_{7}) = (1/2, 0)}.

[figure: y = y_{1}; x = x_{1}; x_{3}y_{3}^{−1}; x_{3}^{−1}y_{3}^{2}; y_{3}; x_{3}y_{3}^{−2}; D_{0}(t); D′_{0}(t); D_{1}(t); D_{2}(t); D_{3}(t); D_{4}(t); D_{5}(t); D_{6}(t); D_{7}(t); 2; 4; 3; 6; 5; 4; 3; 2; 1; v; u; Figure 9.]

Let Y = 2D_{0} + 4D′_{0} + 3D_{1} + 6D_{2} + 5D_{3} + 4D_{4} + 3D_{5} + 2D_{6} + D_{7}, and D = Y_{red}. For simplicity, we rewrite (x_{1}, y_{1}) as (x, y), and set U = Spec C[x, y]. Note that

(U; (x, y)) ⊂ S − D.

Now we consider the curve D_{8}. Let us take the coordinate system (u, v) around the point at infinity of D_{8}. Let U′ be the coordinate system (U′′; (u, v)) ≅ C^{2}. Note that D_{8} ∩ U′ = {u = 0}. By considering the coordinate transformations via above eight blowings-up, we have the following coordinate transformation between (U; (x, y)) and (U′; (u, v)):

u = x^{15}y^{−8} − \frac{1}{4}x^{12}y^{−6} − \frac{t}{2}x^{4}y^{−2} + \frac{1}{2}x^{2}y^{−1},  (5)
v = −x^{−2}y.  (6)

By calculating exterior derivations of u and v, we have

du ∧ dv = x^{12}y^{−8}dx ∧ dy.

On the other hand, by solving the system of equations (5) and (6), we have

x = A^{−1}v^{−2},
y = −A^{−2}v^{−3},

where

A = uv^{6} + \frac{1}{2}v^{5} + \frac{t}{2}v^{4} + \frac{1}{4}.

So,

x^{12}y^{−8} = A^{4} = (uv^{6} + \frac{1}{2}v^{5} + \frac{t}{2}v^{4} + \frac{1}{4})^{4}.

Therefore, we have

dx ∧ dy = \frac{du ∧ dv}{(uv^{6} + \frac{1}{2}v^{5} + \frac{t}{2}v^{4} + \frac{1}{4})^{4}} .

On the other hand, note that we are denoting (x_{1}, y_{1}) by (x, y). By using x_{1} = x_{3}^{−1} and y_{1} = −x_{3}^{−2}y_{3} (see (4)), from (5) and (6), we see that

u = x_{3}y_{3}^{−8} − \frac{1}{4}y_{3}^{−6} − \frac{t}{2}y_{3}^{−2} − \frac{1}{2}y_{3}^{−1},
v = y_{3}.

Then we have

uv^{6} + \frac{1}{2}v^{5} + \frac{t}{2}v^{4} + \frac{1}{4} = x_{3}y_{3}^{−2}.

From Figure 9, the curve C = {uv^{6}+(1/2)v^{5}+(t/2)v^{4}+(1/4) = 0} coincides with the component D′_{0}(t) of the divisor Y. Note that the order of the pole at C of the 2-form dx_{0} ∧ dy_{0} and the multiplicity of D′_{0}(t) in Y are both equal to four. Namely the affine chart (U′; (u, v)) ≅ C^{2} intersects Y, and satisfies that U′ ∩ Y = D′_{0}(t).

Next, let us perform the third blowing-up (the blowing-up with center of a_{2}(t)), by using the coordinate system whose origin is Q(t), namely (z_{2}, W_{2}). For simplicity, let us use the same notations (Z_{i}, w_{i}), (z_{i}, W_{i}), or D_{i} as above. But we denote (u′, v′) for the coordinate system around the point at infinity of D_{8}. And let U′′ be the coordinate system (U′′; (u′, v′)) ≅ C^{2}. Now we have the following results:

u′ = x^{−9}y^{8} − 4x^{−6}y^{6} − \frac{t}{2}x^{−2}y^{2} + \frac{1}{2}x^{−1}y
= x_{3}^{−7}y_{3}^{8} − 4x_{3}^{−6}y_{3}^{6} − \frac{t}{2}x_{3}^{−2}y_{3}^{2} − \frac{1}{2}x_{3}^{−1}y_{3}
v′ = −xy^{−1}
= −x_{3}y_{3}^{−1}

By the similar calculation, we have

dx ∧ dy = dx_{1} ∧ dy_{1} = \frac{du′ ∧ dv′}{(u′(v′)^{6} + \frac{1}{2}(v′)^{5} + \frac{t}{2}(v′)^{4} + 4)^{3}} ,

and

u′(v′)^{6} + \frac{1}{2}(v′)^{5} + \frac{t}{2}(v′)^{4} + 4 = x_{3}^{−1}y_{3}^{2}.

From Figure 9, the curve C′ = {u′(v′)^{6} +(1/2)(v′)^{5} +(t/2)(v′)^{4} +4 = 0} coincides with the component D_{1}(t) of the divisor Y. The order of the pole C′ of the 2-form dx_{1} ∧ dy_{1} and the multiplicity of D_{1}(t) in Y are both equal to three. We see that U′′ ∩ Y = D′_{0}(t).

Consequently, for an Okamoto–Painlevé pair of type Ẽ_{8}, S − Y is covered by (U; (x_{1}, y_{1})) ≅ C^{2} and (U′; (u, v)) − C ≅ C^{2} − C (or (U′′; (u′, v′)) − C′).

3. Okamoto–Painlevé pairs of non-elliptic type

Definition 3.1. An Okamoto–Painlevé pair (S, Y) is of elliptic type if there exists a fibration f : S ⟶ P^{1} of elliptic curves such that a scheme theoretic fiber f^{∗}(∞) at ∞ is Y, that is, f^{∗}(∞) = Y. If (S, Y) is not of elliptic type, we call (S, Y) is of non-elliptic type.

A rational elliptic surface can be obtained by blowings-up of 9 base points of a cubic pencil on P^{2}. Actually, one can obtain an Okamoto–Painlevé pairs (S, Y) of type Ẽ_{6} by blowings-up of infintely near base points of a pencil of 3 × line and a non-singular cubic curve. This example gives an elliptic fibration f : S ⟶ P^{1} with f^{∗}(∞) = Y, and hence (S, Y) is an Okamoto-Painlevé pair of elliptic type. On the other hand, if one blows up 8 (infinitely near) base points of the pencil and blows up a point on anti-canonical divisor which is not a base points of the pencil, one can not obtain an elliptic fibration, so (S, Y) becomes an Okamoto-Painlevé pair of non-elliptic type. For each type of Okamoto-Painlevé pair, one can obtain both elliptic type and non-elliptic type depending on the position of the points of blowings-up and blowings-down. Note that non-elliptic type is general in the moduli space of Okamoto-Painlevé pairs of each type.

The following porposition is shown in [Sa–T].

Proposition 3.1. Let (S, Y) be an Okamoto–Painlevé pair. Then (S, Y) is of non-elliptic type if and only if H^{0}(S − Y, O^{alg}) ≅ C. (Here H^{0}(S − Y, O^{alg}) ≅ C means that all of algebraic regular functions on S − Y are constant functions.)

4. Construction of type D̃_{7}

Now we will construct the Okamoto–Painlevé pair of type D̃_{7} by blowing-up of F_{0} = P^{1} × P^{1} on −K_{F_{0}} = 2s_{0} + 2f. In the following figure two numbers near each solid line denotes the multiplicity and the self-intersection number in −K_{S}. The broken lines denote (−1)-curves, whose multiplicities in F are zero. Namely, they are additional components of F.

[figure: (2); 0; (2); 0; (2); −1; (2); −1; (1); −1; (1); −1; (2); −2; (2); −2; (2); −1; (2); −1; (1); −2; (1); −2]

[figure: (2); −2; (2); −2; (2); −2; (2); −2; (1); −2; (1); −2; (1); −1; (1); −1; (2); −2; (2); −2; (2); −2; (2); −2; (1); −2; (1); −2; (1); −2; (1); −2; −1; −1; Figure 10.]

5. Additional Components

By definition, each Okamoto–Painlevé pair (S, Y) contains the affine plain C^{2} as a Zariski open set and we set

F = S − C^{2}.

We can see F as a divisor with ten components which is obtained by adding some irreducible smooth rational curves to Y_{red}. We call such curves additional components.

Now we will consider F counting the multiplicity m_{j} for the component F_{j} (j = 1, …, 10). We set F = ∑_{j=1}^{10} m_{j}F_{j} and Y = ∑_{i=1}^{r} a_{i}Y_{i}. Note that r < 10. If F_{j} = Y_{i}, then we set m_{j} := a_{i}. And if F_{j} is an additional component, then we set m_{j} = 0. Lemma 5.1. Let C be an additional component of F which intersects F_{j}. Then we have C^{2} = m_{j} −2 Proof. Note that F_{j} which satisfies m_{j} > 0 and which C intersects exists uniquely. By the adjunction formula, we have C^{2} = C · Y − 2. Since C · Y = m_{j}, our assertion is proved.

Now we assume that F = ∑_{i=1}^{10} F_{i} satisfies the following condition.

(∗) Both (F_{1})^{2} and (F_{2})^{2} are even numbers, and both F_{1} and F_{2} intersect only F_{3}.

[figure: F_{1}; F_{2}; F_{3}; Figure 11.]

Lemma 5.2. There exists no Okamoto–Painlevé pairs which satisfy (∗). Proof. We consider the determinant of the intersection matrix of F.

det I_{F} = | \begin{matrix}(F_{1})^{2} & 0 & 1 & 0 & ⋯ & 0 \\ 0 & (F_{2})^{2} & 1 & 0 & ⋯ & 0 \\ 1 & 1 & (F_{3})^{2} & ∗ & ⋯ & ∗ \\ 0 & 0 & ∗ & ∗ & ⋯ & ∗ \\ ⋮ & ⋮ & ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & ∗ & ∗ & ⋯ & ∗ \end{matrix} |

We see that det I_{F} must be an even number, if we expand the determinant. Namely det I_{F} ≠ −1. This contradicts the fact that I_{F} is a unimodular matrix with the signature (b_{+}, b_{−}) = (1, 9). This completes the proof. Lemma 5.3. There is not more than one additional component which intersects the components of F whose multiplicities are ≥ 2.

Proof. We consider additional components C_{1} and C_{2}. We assume that C_{1} and C_{2} intersect F_{1} and F_{2} respectively, and that m_{1} ≥ 2 and m_{2} ≥ 2. Note that C_{1} · C_{2} = 0. If C_{1} · C_{2} ≠ 0, then F includes a cycle. From Lemma 5.1, it follows that (C_{1})^{2} = m_{1} − 2 and (C_{2})^{2} = m_{2} − 2. Note that (C_{1})^{2} ≥ 0 and (C_{2})^{2} ≥ 0. We will prove by separating into some cases.

[i] The case where m_{1} ≥ 3 or m_{2} ≥ 3. For example, we assume m_{1} ≥ 3. We have (C_{1})^{2} > 0 and C_{1} · C_{2} = 0. Therefore, the Hodge index theorem shows that (C_{2})^{2} < 0 or C_{2} ≡ 0. This is a contradiction.

[ii] The case where m_{1} = m_{2} = 2.

(1) The case where another additional component intersects C_{1} or C_{2}. In this case, we suppose that C_{3} intersects C_{1} as an example. Then we have (C_{3})^{2} = −2, C_{1} · C_{3} = 1 and C_{2} · C_{3} = 0. Now we consider a divisor A = nC_{1} + C_{3} for n ∈ Z_{≥2}. We have A^{2} = 2n − 2 > 0 and A · C_{2} = 0. By applying the Hodge index theorem again, it follows that (C_{2})^{2} < 0 or C_{2} ≡ 0. Thus we obtained a contradiction.

(2) The case where another additional component intersect neither C_{1} nor C_{2}. In this case, C_{1} and C_{2} can not intersect the same component of F by Lemma 5.2. Hence we see that C_{1} and C_{2} intersects the different components. Let us denote them by F_{1} and F_{2} respectively. Note that (C_{1})^{2} = (C_{2})^{2} = 0, C_{1} · F_{1} = 1, C_{2} · F_{1} = 0 and (F_{1})^{2} = −2. Now we consider a divisor B = nC_{1} + F_{1} for n ∈ Z_{≥2}. Then we have B^{2} = 2n − 2 > 0 and B · C_{2} = 0. This contradicts the Hodge index theorem. This completes the proof.

By using the above lemmas, we will be able to give the complete list of configurations of F and Y ⊂ F.

Moreover, one can also check that each pattern in the following list can be transformed into (P^{2}, −K_{P^{2}} = 3h), (F_{0}, −K_{F_{0}} = 2s_{0} + 2f) or (F_{2}, −K_{F_{2}} = 2s_{∞}) by performing blowing-up and blowing-down to S on −K_{S}. Since (F_{0}, −K_{F_{0}} = 2s_{0} + 2f) or (F_{2}, −K_{F_{2}} = 2s_{∞}) can be transformed into (P^{2}, −K_{P^{2}} = 3h) by birational transformations, as a consequence, each pattern can be transformed into (P^{2}, −K_{P^{2}} = 3h) by performing blowing-up and blowing-down of (S, Y). (This gives a proof of the assertion (iii) of Theorem 2.1.) This also shows that all of the configuration of F in the following list do exist. In fact, one can perform the inverse birationl transformation from (P^{2}, −3l) to (S, Y) or (S, F).

COMPLETE LIST OF CONFIGURATIONS OF F.

We will give the list of configurations of F in the form of dual graphs. In each figure, we denote by each vertex a component with the positive multiplicity in F, and by each circle an additional component. The number near each vertex means the multiplicity of the component in F, and the number inside each circle means the self-intersection number of the corresponding additional component. Note that the multiplicity of an additional component in F is zero.

[figure: Type D̃_{4}; D̃_{4}–(1); −1; −1; −1; −1; 0; D̃_{4}–(2); −1; −1; −1; −1; −1; 1; 1; 1; 1; 2; 1; 1; 1; 1; 2]

[figure: D̃_{4}–(3); −1; −1; −1; −1; −2; D̃_{4}–(4); −1; −1; −1; 0; −2; 1; 1; 1; 1; 2; 1; 1; 1; 1; 2]

[figure: D̃_{4}–(5); −1; −1; −1; −1; −1; D̃_{4}–(6); −1; −1; −2; −1; −1; 1; 1; 1; 1; 2; 1; 1; 1; 1; 2]

[figure: D̃_{4}–(7); −1; −1; −1; −2; −2; 1; 1; 1; 1; 2]

[figure: Type D̃_{5}; D̃_{5}–(1); −1; −1; −1; −1; 1; 1; 1; 1; 2; 2; D̃_{5}–(2); −1; −1; −1; −1; 1; 1; 1; 1; 2; 2]

[figure: D̃_{5}–(3); −1; −1; −2; −1; 1; 1; 1; 1; 2; 2; D̃_{5}–(4); −1; −1; 0; −1; 1; 1; 1; 1; 2; 2]

[figure: D̃_{5}–(5); −1; −1; 0; −2; 1; 1; 1; 1; 2; 2; D̃_{5}–(6); −1; −1; −1; −1; 1; 1; 1; 1; 2; 2]

[figure: D̃_{5}–(7); −1; −1; −1; −2; 1; 1; 1; 1; 2; 2; D̃_{5}–(8); −1; −1; −2; −2; 1; 1; 1; 1; 2; 2]

[figure: Type D̃_{6}; D̃_{6}–(1); −1; −1; −1; D̃_{6}–(2); −1; −1; 0; 1; 1; 1; 1; 2; 2; 2; 1; 1; 1; 1; 2; 2; 2]

[figure: Type D̃_{7}; −1; −1; 1; 1; 1; 1; 2; 2; 2; 2]

[figure: Type Ẽ_{6}; −1; −1; −1; Ẽ_{6}–(1); −1; −1; 1; Ẽ_{6}–(2); 1; 2; 3; 2; 1; 2; 1; 1; 2; 3; 2; 1; 2; 1]

[figure: −1; −1; 0; Ẽ_{6}–(3); −1; −1; −1; Ẽ_{6}–(4); 1; 2; 3; 2; 1; 2; 1; 1; 2; 3; 2; 1; 2; 1]

[figure: −1; −1; −2; Ẽ_{6}–(5); −1; 0; −2; Ẽ_{6}–(6); 1; 2; 3; 2; 1; 2; 1; 1; 2; 3; 2; 1; 2; 1]

[figure: Type Ẽ_{7}; −1; −1; 1; 2; 3; 4; 3; 2; 1; 2; Ẽ_{7}–(1)]

[figure: −1; 1; 1; 2; 3; 4; 3; 2; 1; 2; Ẽ_{7}–(2)]

[figure: −1; 0; Ẽ_{7}–(3); 1; 2; 3; 4; 3; 2; 1; 2]

[figure: Type Ẽ_{8}; −1; 1; 2; 3; 4; 5; 6; 4; 2; 3]

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[Sak] H. Sakai, Rational surfaces associated with affine root systems and geometry of the Painlevé equations, Kyoto-Math 99-10, preprint(1999).

[ST] T. Shioda and K. Takano, On Some Hamiltonian Structures of Painlevé Systems, I, Funkcial. Ekvac. 40 (1997), 271–291.

[Tak] T. Takebe, Classification of Okamoto–Painlevé pairs, Master thesis, Kobe University (1999), February.

Department of Mathematics, Faculty of Science, Kobe Unversity, Kobe, Rokko, 657-8501, Japan.

   E-mail address: mhsaito@math.kobe-u.ac.jp
   E-mail address: takebe@math.kobe-u.ac.jp