Finite branch solutions to Painleve VI around a fixed singular point
Finite Branch Solutions to Painlevé VI
Around a Fixed Singular Point∗
Katsunori Iwasaki
Faculty of Mathematics, Kyushu University
6-10-1 Hakozaki, Higashi-ku, Fukuoka 812-8581 Japan†
Dedicated to Professor Kazuo Okamoto on his sixtieth birthday
Abstract
Every finite branch local solution to the sixth Painlevé equation around a fixed singular
point is an algebraic branch solution. In particular a global solution is an algebraic
solution if and only if it is finitely many-valued globally. The proof of this result relies
on algebraic geometry of Painlevé VI, Riemann-Hilbert correspondence, geometry and
dynamics on cubic surfaces, resolutions of Kleinian singularities, and power geometry of
algebraic differential equations. In the course of the proof we are also able to classify all
finite branch solutions up to Bäcklund transformations.
1 Introduction
We are interested in a finite branch local solution to the sixth Painlevé equation around a fixed
singular point. We show that every such solution is in fact an algebraic branch solution (see
Definition 1.1 for the terminology). In particular a global solution is an algebraic solution if and
only if it is finitely many-valued globally. Although the problem under study is local in nature,
our solution to it relies on an effective combination of some global technologies and some local
tools. The former includes the algebraic geometry of the sixth Painlevé equation, Riemann-
Hilbert correspondence, geometry and dynamics on cubic surfaces, Kleinian singularities and
their minimal resolutions [15, 16, 17, 18, 20], while the latter includes the power geometry
of algebraic differential equation [5, 6, 7], which is a method of constructing formal solutions
by means of Newton polygons, and the theory of nonlinear differential equations of “regular
singular type” [10, 11], which discusses the convergence of formal solutions.
Let us describe our main results in more detail. First we recall that the sixth Painlevé
equation PVI(κ) is a Hamiltonian system of nonlinear differential equations
∂H(κ)
= −∂H(κ)
, (1)
∗Mathematics Subject Classification: 34M55, 37F10.
†E-mail address: iwasaki@math.kyushu-u.ac.jp
http://arxiv.org/abs/0704.0679v1
with time variable z ∈ Z := P1 − {0, 1,∞} and unknown functions q = q(z) and p = p(z),
depending on complex parameters κ = (κ0, κ1, κ2, κ3, κ4) in the 4-dimensional affine space
K := { κ = (κ0, κ1, κ2, κ3, κ4) ∈ C5 : 2κ0 + κ1 + κ2 + κ3 + κ4 = 1 }, (2)
where the Hamiltonian H(κ) = H(q, p, z; κ) is given by
z(z − 1)H(κ) = (q0q1qz)p2 − {κ1q1qz + (κ2 − 1)q0q1 + κ3q0qz}p+ κ0(κ0 + κ4)qz,
with qν := q − ν for ν ∈ {0, 1, z}. Each of the points 0, 1, ∞ is called a fixed singular point.
It is well known that equation (1) has the analytic Painlevé property, that is, any meromor-
phic solution germ at a base point z ∈ Z can be continued meromorphically along any path in
Z emanating from z. Thus a solution can branch only around a fixed singular point. We are
interested in finite branch solutions around it, by which we mean the following.
Definition 1.1 A finite branch solution to equation (1), say, around z = 0 is a local solution
(q(z), p(z)) on a punctured disk D× = D − {0} centered at z = 0 such that its lift (q̃(z̃), p̃(z̃))
along some finite branched covering ϕ : (D̃, 0̃) → (D, 0), z̃ 7→ z = z̃n around z = 0 is a single-
valued meromorphic function on D̃× = D̃ − {0̃}. Such a solution is said to be an algebraic
branch solution if it can be represented by a convergent Puiseux-Laurent expansion
q(z) =
i/n, p(z) =
i/n, (3)
with ai = bi = 0 for all sufficiently small i ≪ 0, namely, if the lift (q̃(z̃), p̃(z̃)) is a single-valued
meromorphic function on D̃ with at most pole at the origin z̃ = 0̃.
Problem 1.2 Is any finite branch solution to PVI(κ) an algebraic branch solution ?
In this article we settle this problem in the affirmative as is stated in the following.
Theorem 1.3 Any finite branch solution to Painlevé VI around a fixed singular point is an
algebraic branch solution. In particular a global solution is an algebraic solution if and only if
it is finitely many-valued globally. These results are valid for all parameters κ ∈ K.
It is an interesting problem to consider algebraic solutions to Painlevé VI. Many algebraic
solutions have been constructed in [1, 2, 3, 8, 13, 14, 24, 25], but a complete classification seems
to be outstanding. We hope that Theorem 1.3 will play an important part in discussing this
issue. The following remark explains what Theorem 1.3 signifies and why it is remarkable.
Remark 1.4 Logically, according to Definition 1.1, a finite branch solution (q(z), p(z)) around
z = 0 may have a very transcendental singularity at z = 0, to the effect that its lift (q̃(z̃), p̃(z̃))
may have infinitely many poles in D̃× accumulating to the origin z̃ = 0̃, or even if such an
accumulation phenomenon does not occur, it may have an essential singularity at z̃ = 0̃.
Rather surprisingly, however, Theorem 1.3 excludes the possibility for a finite branch solution
to admit such transcendental phenomena. This result becomes more intriguing if we recall that
wild behaviors of a generic solution to Painlevé VI have been observed in [9, 12, 22, 31, 32] and
examples of solutions with infintely many poles accumulating to z̃ = 0 are given in [12, 31];
such a distribution of poles may be expected for a generic solution, though it is not rigorously
verified yet to the author’s knowledge. Thus we can think that a finite branch solution is quite
distinguished from generic solutions, necessarily being an algebraic branch solution.
I II III
{all algebraic branch solutions} i→֒ {all finite branch solutions} j→֒ {all solutions}
{some algebraic branch solutions} i
→֒ {all finite branch solutions}
I ′ II
Figure 1: Main idea for the proof of Theorem 1.3
The main idea for the proof of Theorem 1.3 is presented in Figure 1. We have natural
inclusions i and j in the top line of Figure 1 and we wish to show that the injection i is in fact
a surjection. Our strategy consists of the “upper bound part” and the “lower bound part”.
(1) Upper bound part: In this part we investigate the inclusion j : II →֒ III in Figure 1,
considering how the locus of finite branch solutions is included in the moduli space of
all solutions. In other words, we make a confinement of the locus II in the entire space
III. What we shall really do is not an upper bound estimation of this locus but rather a
pinpoint identification of it. This is the main part of the article and we use the algebraic
geometry of Painlevé VI, Riemann-Hilbert correspondence, geometry and dynamics on
cubic surfaces, and minimal resolutions of Kleinian singularities [15, 16, 17, 18, 19, 20].
(2) Lower bound part: In this part we fill in the diagram of Figure 1 by adding the bottom
line to the top one. We try to construct as many algebraic branch solutions as possible
in order to make the set I ′ as large as possible. The construction is based on the power
geometry technique developed in [5, 6, 7] and the convergence arguments in [10, 11].
We are done if the set I ′ is large enough to show that the injection i′ : I ′ →֒ II is in
fact a surjection. This does not mean that we verify the equality I ′ = II directly. (If
such a direct approach were feasible, then our problem would not be difficult from the
beginning!) Instead, we prove it very indirectly based on the following idea.
(3) Key trick: Suppose that a component A of I ′ injects into a component B of II. If the
cardinalities of A and B are finite and the same, then the injection i′ : A →֒ B is in
fact a surjection. If A and B are biholomorphic to C and the injection i′ : A →֒ B is
holomorphic, then it must be a surjection because any holomorphic injection C →֒ C is a
surjection (use Casorati-Weierstrass or Picard’s little theorem). The same argument holds
true if C is replaced by C×, since any holomorphic injection C× →֒ C× is a surjection (lift
it to the universal covering C →֒ C). These tricks enable us to identify the component
A ⊂ I ′ with the component B ⊂ II. We show that each component involved is either of
the three types mentioned above. Then we make this kind of argument componentwise
to get an identification I ′ = II, which leads to the desired coincidence I ′ = I = II.
In view of the way in which Theorem 1.3 is established, the power geometry technique
provides us with an efficient method of identifying all finite branch solutions (up to Bäcklund
transformations), which have now turned out to be algebraic branch solutions, by determining
the leading terms of their Puiseux-Laurent expansions.
In some sense this article is a counterpart of the previous paper [20] where an ergodic study
of Painlevé VI is developed (see also the survey [21]). Put z1 = 0, z2 = 1, z3 = ∞. For each
γ2 γ3
Figure 2: Three basic loops γ1, γ2, γ3 in Z = P
1 − {0, 1,∞}
{i, j, k} = {1, 2, 3}, let γi be a loop in Z surrounding zi once anti-clockwise and leaving zj and
zk outside as in Figure 2. Then the fundamental group π1(Z, z) is represented as
π1(Z, z) = 〈 γ1, γ2, γ3 | γ1γ2γ3 = 1 〉. (4)
A loop γ ∈ π1(Z, z) is said to be elementary if it is conjugate to γmi for some i ∈ {1, 2, 3} and
m ∈ Z; otherwise, it is said to be non-elementary. The main theme of [20] is the dynamics of
the nonlinear monodromy of PVI(κ) along a given loop γ. It is shown there that, along every
non-elementary loop, the nonlinear monodromy is chaotic and the number of its periodic points
grows exponentially as the period tends to infinity. On the other hand, it is Liouville integrable
along an elementary loop, in the sense that it preserves a Lagrangian fibration. Now we notice
that from the dynamical point of view the main problem of this article is nothing other than
discussing the periodic points of the nonlinear monodromy along the basic loop γi, which is
of course an elementary loop. In view of its integrable character, one may doubt if there is
something very deep with this issue. As Theorem 1.3 and Remark 1.4 show, however, this issue
is actually quite interesting from the function-theoretical point of view.
The plan of this article is as follows. In §2 the phase space of Painlevé VI is introduced as a
moduli space of stable parabolic connections. In §3 the Riemann-Hilbert correspondence from
the moduli space to an affine cubic surface is formulated and its character as an analytic minimal
resolution of Kleininan singularities is stated. In §4 the dynamical system on the cubic surface
representing the nonlinear monodromy of Painlevé VI is formulated and some preliminary
properties of it are given. In §5 we briefly review Bäcklund transformations and their relation
to the Riemann-Hilbert correspondence. In §6 fixed points and periodic points of the dynamical
system are discussed. A stratification of the parameter space K is also introduced in order to
describe the singularities of the cubic surfaces. In §7 a case-by-case study of fixed points and
periodic points is made according to the stratification, thereby a pinpoint identification of
finite branch solutions is made on each stratum. In §8 power geometry of algebraic differential
equations is applied to Painlevé VI in order to construct as many algebraic branch solutions as
possible. In §9 we consider the inclusion of those solutions constructed in §8 into the moduli
space of all finite branch solutions. After some preliminaries on Riccati solutions, we show that
this inclusion is in fact a surjection, thereby complete the proof of Theorem 1.3.
singularities t1 = 0 t2 = z t3 = 1 t4 = ∞
first exponent −λ1 −λ2 −λ3 −λ4
second exponent λ1 λ2 λ3 λ4 − 1
difference κ1 κ2 κ3 κ4
Table 1: Riemann scheme: κi is the difference of the second exponent from the first.
2 Phase Space
Equation (1) is only a fragmentary appearance of a more intrinsic object constructed algebro-
geometrically [16, 17, 18]. We review this construction following the expositions of [20, 21]. The
sixth Painlevé dynamical system PVI(κ) is formulated as a holomorphic, uniform, transversal
foliation on a fibration of certain smooth quasi-projective rational surfaces
πκ : M(κ) → Z := P1 − {0, 1,∞},
whose fiber Mz(κ) := π−1κ (z) over z ∈ Z, called the space of initial conditions at time z, is
realized as a moduli space of stable parabolic connections. The total space M(κ) is called
the phase space of PVI(κ). In this formulation, the uniformity of the Painlevé foliation, in
other words, the geometric Painlevé property of it is a natural consequence of a solution to
the Riemann-Hilbert problem (see Theorem 3.5), especially of the properness of the Riemann-
Hilbert correspondence [16]. Then equation (1) is just a coordinate expression of the foliation
on an affine open subset of M(κ) and the analytic Painlevé property for equation (1) is an
immediate consequence of the geometric Painlevé property for the foliation and the algebraicity
of the phase space M(κ). Moreover there exists a natural compactification Mz(κ) →֒ Mz(κ)
of the moduli space Mz(κ) into a moduli space Mz(κ) of stable parabolic phi-connections.
Here we include a very sketchy explanation of the terminology used in the last paragraph.
A stable parabolic connection is a Fuchsian connection equipped with a parabolic structure on
a (rank 2) vector bundle over P1 having a Riemann scheme as in Table 1, where the parabolic
structure corresponds to the first exponents, which satisfies a sort of stability condition in
geometric invariant theory. Here the parameter κi stands for the difference of the second
exponent from the first one at the regular singular point ti. On the other hand, a stable
parabolic phi-connection is a variant of stable parabolic connection allowing a “matrix-valued
Planck constant” called a phi-operator φ such that the generalized Leibniz rule
∇(fs) = df ⊗ φ(s) + f∇(s)
is satisfied, where the key point here is that the field φ may be degenerate or simi-classical.
Then the moduli space Mz(κ) can be compactified by adding some semi-classical objects, that
is, some stable parabolic phi-connections with degenerate phi-operator φ. There is the following
characterization of our moduli spaces (see Figure 3).
Theorem 2.1 ([16, 17, 18])
(1) The compactified moduli space Mz(κ) is isomorphic to an 8-point blow-up of the Hirze-
bruch surface Σ2 → P1 of degree 2.
Mz(κ)
Yz(κ) : vertical leaves
E1 E2 E3 E4
Figure 3: Nonlinear monodromy γ∗ : Mz(κ) 	 along a loop γ ∈ π1(Z, z)
(2) Mz(κ) has a unique effective anti-canonical divisor Yz(κ), which is given by
Yz(κ) = 2E0 + E1 + E2 + E3 + E4, (5)
where E0 is the strict transform of the section at infinity and Ei (i = 1, 2, 3, 4) is the strict
transform of the fiber over the point ti ∈ P1 of the Hirzebruch surface Σ2 → P1.
(3) The support of the divisor Yz(κ) is exactly the locus where the phi-operator φ is degenerate,
with the coefficients of formula (5) being the ranks of degeneracy of φ. In particular,
Mz(κ) = Mz(κ)− Yz(κ).
This theorem implies that Mz(κ) is a moduli-theoretical realization of the space of initial
conditions for PVI(κ) constructed “by hands” in [26], Mz(κ) is a generalized Halphen surface
of type D
4 in [30] and (Mz(κ),Yz(κ)) is an Okamoto-Painlevé pair of type D̃4 in [28].
Since the Painlevé foliation has the geometric Painlevé property [16], each loop γ ∈ π1(Z, z)
admits global horizontal lifts along the foliation and induces an automorphism
γ∗ : Mz(κ) → Mz(κ), Q 7→ Q′, (6)
called the nonlinear monodromy along the loop γ (see Figure 3). Note that a fixed point or a
periodic point of the map γ∗ : Mz(κ) 	 can be identified with a solution germ at z which is
single-valued or finitely many-valued along the loop γ, respectively.
3 Riemann-Hilbert Correspondence
Generally speaking, a Riemann-Hilbert correspondence is the map from a moduli space of
flat connections to a moduli space of monodromy representations, sending a connection to its
C1 C2 C3
0 z 1
Figure 4: Four loops in P1 − {0, z, 1,∞}
monodromy. In our situation an appropriate Riemann-Hilbert correspondence
RHz,κ : Mz(κ) → Rz(a), Q 7→ ρ, (7)
is formulated in [16, 17, 18]. For each a = (a1, a2, a3, a4) ∈ A := C4, let Rz(a) denote the
moduli space of Jordan equivalence classes of linear monodromy representations
ρ : π1(P
1 − {0, z, 1,∞}, ∗) → SL2(C),
with the prescribed local monodromy data Tr ρ(Ci) = ai (i = 1, 2, 3, 4), where Ci is a loop as in
Figure 4. Any stable parabolic connection Q ∈ Mz(κ), restricted to P1 − {0, z, 1,∞}, induces
a flat connection and determines the Jordan equivalence class ρ ∈ Rz(a) of its monodromy
representations, where the correspondence of parameters κ 7→ a is described as follows. If
−1πκi) (i = 0, 1, 2, 3),
− exp(
−1πκ4) (i = 4),
then b = (b0, b1, b2, b3, b4) belongs to the multiplicative space
B := { b = (b0, b1, b2, b3, b4) ∈ (C×)5 : b20b1b2b3b4 = 1 }.
The Riemann scheme in Table 1 then implies that the monodromy matrix ρ(Ci) has an eigen-
value bi for each i = 1, 2, 3, 4. Since ρ(Ci) ∈ SL2(C), its trace ai = Tr ρ(Ci) is given by
ai = bi + b
i (i = 1, 2, 3, 4). (9)
Given any θ = (θ1, θ2, θ3, θ4) ∈ Θ := C4θ, consider the affine cubic surface
S(θ) = {x ∈ C3x : f(x, θ) := x1x2x3 + x21 + x22 + x23 − θ1x1 − θ2x2 − θ3x3 + θ4 = 0}.
Then there exists an isomorphism of affine algebraic surfaces
Rz(a) → S(θ), ρ 7→ x = (x1, x2, x3), with xi = Tr ρ(CjCk)
for {i, j, k} = {1, 2, 3}, where the correspondence of parameters a 7→ θ is given by
aia4 + ajak ({i, j, k} = {1, 2, 3}),
a1a2a3a4 + a
1 + a
2 + a
3 + a
4 − 4 (i = 4).
w1 w2
w3 w4

−2 1 1 1 1
1 −2 0 0 0
1 0 −2 0 0
1 0 0 −2 0
1 0 0 0 −2

Figure 5: Dynkin diagram and Cartan matrix of type D
The composition of the sequence κ 7→ b 7→ a 7→ θ of the three maps (8), (9) and (10) is referred
to as the Riemann-Hilbert correspondence in the parameter level [16] and is denoted by
rh : K → Θ. (11)
Then the Riemann-Hilbert correspondence (7) is reformulated as a holomorphic map
RHz,κ : Mz(κ) → S(θ) with θ = rh(κ). (12)
The map (11) admits a remarkable affine Weyl group structure [16, 19], from which the
Bäcklund transformations of Painlevé VI emerge [15]. In view of formula (2) the affine space
K can be identified with the linear space C4 by the forgetful isomorphism K → C4, κ =
(κ0, κ1, κ2, κ3, κ4) 7→ (κ1, κ2, κ3, κ4), where the latter space C4 is equipped with the standard
(complex) Euclidean inner product. For each i ∈ {0, 1, 2, 3, 4}, let wi : K → K, κ 7→ κ′, be the
orthogonal reflection in the hyperplane { κ ∈ K : κi = 0}, which is explicitly represented as
κ′j = κj + κicij (i, j ∈ {0, 1, 2, 3, 4}), (13)
where C = (cij) is the Cartan matrix of type D
4 given in Figure 5. Then the group generated
by w0, w1, w2, w3, w4 is an affine Weyl group of type D
4 ) = 〈w0, w1, w2, w3, w4〉 y K.
corresponding to the Dynkin diagram in Figure 5. The reflecting hyperplanes of all reflections
in the group W (D
4 ) are given by affine linear relations
κi = m, κ1 ± κ2 ± κ3 ± κ4 = 2m+ 1 (i ∈ {1, 2, 3, 4}, m ∈ Z),
where the signs ± may be chosen arbitrarily. Let Wall be the union of all these hyperplanes.
Then the affine Weyl group structure on (11) is stated as follows [16] (see Figure 6).
Lemma 3.1 In terms of b ∈ B, the discriminant ∆(θ) of the cubic surfaces S(θ) factors as
∆(θ) =
(bl − b−1l )2
ε∈{±1}4
(bε − 1), (14)
where we put bε = bε11 b
4 for each quadruple sign ε = (ε1, ε2, ε3, ε4) ∈ {±1}4. The
Riemann-Hilbert correspondence in the parameter level (11) is a branched W (D
4 )-covering
ramifying along Wall and mapping Wall onto the discriminant locus ∆(θ) = 0 in Θ.
∆(θ) = 0
K-space Θ-spaceWall
Figure 6: Riemann-Hilbert correspondence in the parameter level
The singularity structure of the cubic surfaces S(θ) can be described in terms of the strati-
fication of K by proper Dynkin subdiagrams, which we now define.
Definition 3.2 Let I be the set of all proper subsets of {0, 1, 2, 3, 4} including the empty set
∅. For each element I ∈ I, we put
KI = the W (D(1)4 )-translates of the set { κ ∈ K : κi = 0 (i ∈ I) },
DI = the Dynkin subdiagram of D
4 that has nodes • exactly in I.
Let KI be the set obtained from KI by removing the sets KJ with #J = #I + 1. Then it
turns out that we have either KI = KI′ or KI ∩ KI′ = ∅ for any distinct subsets I, I ′ ∈ I
(see Remark 3.3). So we can think of the stratification of K by the subsets KI (I ∈ I), called
the W (D
4 )-stratification, where each KI is referred to as a W (D
4 )-stratum. For example, if
I = ∅, one has the big open K∅ = K −Wall. Other examples of W (D()14 )-strata are given in
Figure 7. The diagram DI encodes not only its underlying abstract Dynkin type but also the
inclusion pattern DI →֒ D(1)4 , a kind of marking. The abstact Dynkin type of DI is denoted by
Dynk(I). All the feasible abstract Dynkin types are tabulated in Table 2.
There is a mistake in the definition of KI in [16, Definition 9.3] and [21], which is now
corrected in Definition 3.2. (As for [16], correction may be possible before it is published.)
Remark 3.3 Let I and I ′ be distinct elements of I. If Dynk(I) 6= Dynk(I ′), then KI∩KI′ = ∅.
On the other hand, if KI = KI′ then Dynk(I) = Dynk(I ′) must be of abstract type A1 or A2.
D4 A⊕41 A3
I = {0, 1, 2, 3} I = {1, 2, 3, 4} I = {0, 1, 2}
Figure 7: Examples of W (D
4 )-strata
number of nodes 4 3 2 1 0
abstract D4 A3 A2 A1 ∅
Dynkin type A⊕41 A
1 − −
Table 2: Feasible abstract Dynkin types
(1) There is a unique W (D
4 )-stratum of abstract type ∅, or A1, or A2, or A⊕41 .
(2) There are six W (D
4 )-strata of abstract type A
1 or A3.
(3) There are four W (D
4 )-strata of abstract type A
1 or D4.
Example 3.4 We consider the W (D
4 )-strata of abstract types A
1 and D4.
(1) The unique W (D
4 )-stratum of abstract type A
1 exactly corresponds to the value θ =
(0, 0, 0,−4). A parameter κ ∈ K lies in this stratum if and only if either
(a) κ1, κ2, κ3, κ4 ∈ Z, κ1 + κ2 + κ3 + κ4 ∈ 2Z; or
(b) κ1, κ2, κ3, κ4 ∈ Z+ 1/2.
(2) The four W (D
4 )-strata of abstract type D4 exactly correspond to the values θ =
(8ε1, 8ε2, 8ε3, 28), where ε = (ε1, ε2, ε3) ∈ {±}3 ranges over all triple signs such that
ε1ε2ε3 = 1. A parameter κ ∈ K lies in the union of these W (D(1)4 )-strata if and only if
κ1, κ2, κ3, κ4 ∈ Z, κ1 + κ2 + κ3 + κ4 ∈ 2Z+ 1.
With this stratification, we have a very neat solution to the Riemann-Hilbert problem.
Theorem 3.5 ([16, 17, 18]) Given any κ ∈ K, put θ = rh(κ) ∈ Θ. Then,
(1) if κ ∈ KI then S(θ) has Kleinian singularities of Dynkin type DI ,
(2) the Riemann-Hilbert correspondence (12) is a proper surjective map that is an analytic
minimal resolution of Kleinian singularities.
If κ ∈ K −Wall then the surface S(θ) is smooth and RHz,κ is a biholomorphism, while if
κ ∈ Wall, it is not a biholomorphism but only gives a resolution of singularities (proper and
surjective, but not injective). For example, see Figure 8 for the case κ = (0, 0, 0, 0, 1) where
a singularity of type D4 occurs. In the latter case, however, if we take a standard algebraic
minimal resolution of Kleinian singularities as constructed by Brieskorn [4] and others,
ϕ : S̃(θ) → S(θ) (15)
moduli space cubic surface
RHz,κ
resolution of
singularity
Mz(κ)
D4Ez(κ)
Figure 8: Resolution of singularities by Riemann-Hilbert correspondence
then we can lift the Riemann-Hilbert correspondence (12) to have a commutative diagram
Mz(κ)
gRHz,κ−−−→ S̃(θ)
Mz(κ)
RHz,κ−−−→ S(θ).
The lifted Riemann-Hilbert correspondence R̃Hz,κ is a biholomorphism and hence gives a strict
conjugacy between the nonlinear monodromy (6) of PVI(κ) and a certain automorphism
g̃ : S̃(θ) → S̃(θ). (17)
This latter map will be described explicitly in Section 4 (see Theorem 4.1).
The singularity structure of the affine cubic surface S(θ) is closely related to the Riccati
solutions to PVI(κ) [16], where a Riccati solution is a particular solution that arises from the
Riccati equation associated to a Gauss hypergeometric equation. Let Ez(κ) ⊂ Mz(κ) be the
exceptional set of the resolution of singularities by the Riemann-Hilbert correspondence (12).
Similarly, let E(θ) ⊂ S̃(θ) be the exceptional set of the algebraic resolution of singularities (15).
Theorem 3.6 ([16, 29]) Equation PVI(κ) admits Riccati solutions if and only if κ ∈ Wall.
All Riccati solution germs at time z ∈ Z are parametrized by the exceptional set Ez(κ) ⊂ Mz(κ),
which precisely corresponds to the exceptional set E(θ) ⊂ S̃(θ) through the lifted Riemann-Hilbert
correspondence (16).
Fot this reason we may refer to Ez(κ) and M◦z(κ) := Mz(κ) − Ez(κ) as the Riccati locus and
non-Riccati locus of Mz(κ) respectively. They are invariant under the action of the nonlinear
monodromy (6). Corresponding to them, let Sing(θ) and S◦(θ) := S(θ)−Sing(θ) be the singular
locus and the smooth locus of the cubic surface S(θ) respectively.
Remark 3.7 Two remarks are in order at this stage.
(1) By Theorem 3.5 the Riemann-Hilbert correspondence (12) restricts to a biholomorphism
RH◦z,κ : M◦z(κ) → S◦(θ) (18)
between the non-Riccati locus of Mz(κ) and the smooth locus of S(θ), while it collapses
the Riccati locus Ez(κ) to the singular locus Sing(θ). In order to resolve this degeneracy
and obtain an isomorphism, we had to take the lifted Riemann-Hibert correspondence
(16), which induces an isomorphism between the exceptional sets Ez(κ) and E(θ).
(2) For the Riccati solutions the main problem of this article is trivial; if a Riccati solution
is a finite branch solution around a fixed singular point, then it is an algebraic branch
solution, because the Riccati solution is (essentially) the logarithmic derivative of a Gauss
hypergeometric function. Thus we may restrict our attention to the non-Riccati locus.
4 Dynamics on Cubic Surface
We shall describe the strict conjugacy (17) of the nonlinear monodromy (6). For a cyclic
permutation (i, j, k) of (1, 2, 3) we define an isomorphism gi : S(θ) → S(θ′), (x, θ) 7→ (x′, θ′) by
gi : (x
j , x
j , θ
k) = (θj − xj − xkxi, xi, xk, θj , θi, θk). (19)
Through the resolution of singularities (15), the map gi is uniquely lifted to an isomorphism
g̃i : S̃(θ) → S̃(θ′), (i = 1, 2, 3).
We remark that the square g2i is an automorphism of S(θ) with g̃2i being its lift to S̃(θ).
Theorem 4.1 ([16]) For each i ∈ {1, 2, 3} the nonlinear monodromy γi∗ : Mz(κ) 	 along the
i-th basic loop γi is strictly conjugated to the automorphism g̃
i : S̃(θ) 	 via the lifted Riemann-
Hilbert correspondence (16). More generally, if γ ∈ π1(Z, z) is represented by γ = γε1i1 γ
· · ·γεnin
with (i1, . . . , in) ∈ {1, 2, 3}n and (ε1, . . . , εn) ∈ {±1}n, then the map (17) is given by
g̃ = g̃2ε1i1 g̃
· · · g̃2εnin .
Let F̃ixj(θ) be the set of all fixed points of the transformation g̃
j : S̃(θ) 	. Moreover,
for any integer n > 1, let P̃erj(θ;n) be the set of all periodic points of prime period n of the
transformation g̃2j : S̃(θ) 	. Theorem 4.1 then implies that all single-valued solution germs and
all n-branch solution germs to PVI(κ) around the fixed singular point zj are parametrized by
the sets F̃ixj(θ) and P̃erj(θ;n) respectively. By Remark 3.7, considering F̃ixj(θ) and P̃erj(θ;n)
upstairs is the same thing as considering Fixj(θ) and Perj(θ;n) downstairs, except for the ex-
ceptional locus upstairs and the singular locus downstairs. Here Fixj(θ) and Perj(θ;n) denote
the set of all fixed points and the set of all periodic points of prime period n of the transfor-
mation g2j : S(θ) 	 downstairs. In order to make the situation more transparent, we begin by
investigating simultaneous fixed points of g21, g
3 downstairs.
Theorem 4.2 If Fix(θ) is the set of all simultaneous fixed points of g21, g
3 : S(θ) 	, then
Fix(θ) = Sing(θ). (20)
Proof. A point x ∈ S(θ) is a singular point of the surface S(θ) if and only if its gradient vector
field y(x, θ) = (y1(x, θ), y2(x, θ), y3(x, θ)) vanishes at the point x, where
yi(x, θ) :=
(x, θ) = 2xi + xjxk − θi (21)
On the other hand, an inspection of formula (19) readily shows that x ∈ S(θ) is a simultaneous
fixed point of g21, g
3 if and only if x is a common root of equations
f(x, θ) = y1(x, θ) = y2(x, θ) = y3(x, θ) = 0. (22)
Then the equality (20) immediately follows from these observations. ✷
As is announced in [16], this theorem yields a characterization of the rational solutions.
Corollary 4.3 Any single-valued global solution to PVI(κ) is a rational Riccati solution.
Proof. If a single-valued solution Q ∈ Mz(κ) belongs to the non-Riccati locus Q ∈ M◦z(κ),
then the Riemann-Hilbert correspondence (18) sends Q to a smooth point x ∈ S◦(θ). Since the
single-valued solution Q is a simultaneous fixed point of the nonlinear monodromies γ1∗, γ2∗,
γ3∗, the corresponding point x must lie in Fix(θ). Then Theorem 4.2 implies that x ∈ Sing(θ),
which contradicts the fact that x ∈ S◦(θ). Hence any single-valued solution is a Riccati solution.
Since any Riccati solution is (essentially) the logarithmic derivative of a Gauss hypergeometric
function, any single-valued Riccati solution must be a rational solution. ✷
All the rational solutions to Painlevé VI are classified in [25]. We come back to our discussion
downstairs and give a simple characterization of the sets Fixj(θ) and Perj(θ;n).
Lemma 4.4 Let x = (x1, x2, x3) ∈ S(θ) be any point and let n be any integer > 1.
(1) x ∈ Fixj(θ) if and only if x is a root of equations
f(x, θ) = yj(x, θ) = yk(x, θ) = 0. (23)
(2) x ∈ Perj(θ;n) if and only if there exists an integer 0 < m < n coprime to n such that
f(x, θ) = 0, xi = 2 cos(πm/n). (24)
Proof. We put (x′, θ′) = gj(x, θ) and y
′ = y(x′, θ′). Then formula (19) yields
y′i = yi − xjyk, y′j = −yk, y′k = yj − xiyk. (25)
For each integer n ∈ Z we write (x(n), θ(n)) = gni (x, θ) and y(n) = y(x(n), θ(n)). From formulas
(19) and (25), we can easily obtain three recurrence relations
(n+2)
j + xi y
(n+1)
j + y
j = 0, (26)
(n+2)
j − x
j = y
(n+2)
j , (27)
(n+1)
k = x
j . (28)
The characteristic equation of the recurrence relation (26) is the quadratic equation
λ2 + xi λ+ 1 = 0, (29)
the roots of which are denoted by α and β = α−1. Since αβ = 1, we may and shall assume
that |α| ≥ 1 ≥ |β| > 0 in the sequel. The discussion is divided into two cases.
Case xi ∈ C − {±2}: In this case, the roots α and β are distinct and different from ±1
and the recurrence relation (26) is settled as
βn(αyj + yk)− αn(βyj + yk)
α− β .
Then it follows from (27) and (28) that the sequences x
j and x
k are determined as
j = x
(2n+1)
k = p α
2n + q β2n + r1,
(2n+1)
j = x
k = p α
2n+1 + q β2n+1 + r2,
where the constants p, q, r1 and r2 are given by
p = − α
2(βyj + yk)
(α − β)(α2 − 1) , q =
β2(αyj + yk)
(α− β)(β2 − 1) ,
r1 = xj − p− q, r2 = x′j − αp− βq.
Notice that p = q = 0 if and only if x satisfies equations (23). Indeed, the condition p = q = 0
is equivalent to αyj + yk = βyj + yk = 0, which is equivalent to the condition yj = yk = 0,
because the roots α and β are distinct.
Now we assume that x is a root of equations (23). Then (30) implies that the sequence x(n)
is periodic of period two, that is, x is a fixed point of g2j . Next we assume that x is not a root
of equations (23). If x is a periodic point of g2j of prime period n ≥ 1, then (30) yields
j − xj = (α2n − 1)(p− qβ2n) = 0,
k − xk = (α2n − 1)(pα− qβ2n+1) = 0.
Here it cannot happen that p− qβ2n = pα− qβ2n+1 = 0. Indeed, otherwise, we have p = qβ2n
and q(1 − β2) = 0. Since at least one of p and q is nonzero, we have β ∈ {±1} and hence
xi ∈ {±2}, which contradicts the assumption that xi 6∈ {±2}. Therefore, α2n = 1, that is, α
is a primitive 2n-th root of unity. Note that n ≥ 2 since α 6∈ {±1}. Thus there is an integer
0 < m < n comprime to n such that α = exp(πim/n) and so xi = α + α
−1 = 2 cos(πm/n),
which leads to condition (24). Conversely, if condition (24) is satisfied, then it is easy to see
that x is a periodic point of g2j of prime period n.
Case xi ∈ {±2}: In this case we have xi = −2ε for some sign ε ∈ {±1} and hence
equation (29) has a double root α = β = ε. Then the recurrence equation (26) is settled as
j = ε
n{yj − n(yj + εyk)}. If the sequence x(n) is periodic, then so is the sequence y(n)j . This
is the case if and only if yj + εyk = 0. Conversely, if this condition is satisfied, then we have
j = ε
nyj. Substituting this equation into (27) yields
j = x
(2n+1)
k = xj + nyj,
(2n+1)
j = x
(2n+2)
k = x
j + εnyj.
Hence the sequence x(2n) is periodic if and only if yj = yk = 0, namely, if and only if x is a root
of (23). In this case x is a fixed point of g2j . ✷
In order to give the relation between the fixed points upstairs and those downstairs, we put
Fix◦j(θ) := Fixj(θ)− Sing(θ), F̃ix
j (θ) := F̃ixj(θ)− E(θ), F̃ix
j(θ) := F̃ixj(θ) ∩ E(θ).
For the periodic points of prime period n > 1, we define Per◦j(θ;n), P̃er
j(θ;n) and P̃er
j(θ;n) in
a similar manner. Then there exist direct sum decompositions
F̃ixj(θ) = F̃ix
j (θ)∐ F̃ix
j(θ), P̃erj(θ;n) = P̃er
j (θ;n)∐ P̃er
j(θ;n),
where the exceptional components F̃ix
j(θ) and P̃er
j(θ;n) parametrize the single-valued Riccati
solutions and the n-branched Riccati solutions around the fixed singular point zj respectively.
Lemma 4.5 The minimal resolution (15) induces an isomorphism
ϕ : F̃ix
j (θ) → Fix◦j (θ). (32)
For any n > 1 we have Per(θ;n)∩Sing(θ) = ∅, that is, Per◦j(θ;n) = Perj(θ;n), and the minimal
resolution (15) induces an isomorphism
ϕ : P̃er
j (θ;n) → Perj(θ;n) (n > 1). (33)
Proof. The isomorphism (32) is trivial from the definition. The assertion Per(θ;n)∩Sing(θ) = ∅
follows from (20). Then the isomorphism (33) is again trivial from the definition. ✷
The fixed point set and the periodic point set, upstairs or downstairs, will be investigated
more closely in §6. For this purpose it is convenient to consider the symmetric group S4 of
degree 4 acting on K by permuting the entries κ1, κ2, κ3, κ4 of κ ∈ K and fixing κ0. Through
the Riemann-Hilbert correspondence in the parameter level, rh : K → Θ, the action S4 y K
induces an action of S3 ⋉ Kl on Θ, where Kl is Klein’s 4-group realized as the group of even
triple signs, Kl = {ε = (ε1, ε2, ε3) ∈ {±1}3 : ε1ε2ε3 = 1}, acting on Θ by the sign changes
(θ1, θ2, θ3, θ4) 7→ (ε1θ1, ε2θ2, ε3θ3, θ4), while S3 acts on Θ by permuting the entries θ1, θ2, θ3 of
θ ∈ Θ and fixing θ4. This construction defines an isomorphism of groups
S4 ∼= S3 ⋉Kl, σ 7→ (τ, ε), (34)
with respect to which the map rh : K → Θ becomes S4-equivariant. Viewed as a subgroup of
S4, Klein’s 4-group is the permutation group Kl = {1, (14)(23), (24)(31), (34)(12)}.
Let σ ∈ S4 act on x = (x1, x2, x3) in the same manner as it does on (θ1, θ2, θ3). Then the
polynomial f(x, θ) is σ-invariant and hence σ induces an isomorphism of algebraic surfaces,
σ : S(θ) → S(σ(θ)). As for the action g2j : S(θ) 	, we have the commutative diagram
j−−−→ S(θ)
S(σ(θ)) −−−→
S(σ(θ)),
(A3)i (A
Figure 9: W̃ (D
4 )-strata (A3)i and (A
for any element σ ∈ S4 with τ ∈ S3 determined by (34). It induces isomorphisms
σ : Fixj(θ) → Fixτ(j)(σ(θ)), σ : Perj(θ;n) → Perτ(j)(σ(θ);n),
which, via the minimal resolution (15), lift up to isomorphisms
σ̃ : F̃ixj(θ) → F̃ixτ(j)(σ(θ)), σ̃ : P̃erj(θ;n) → P̃erτ(j)(σ(θ);n).
The action of the symmetric group S4 on K mentioned above is just induced from its action
on the index set {0, 1, 2, 3, 4} fixing the element 0, namely, from the realization of S4 as the
automorphism group of the Dynkin diagram D
4 . By taking the semi-direct product by the
symmetric group S4 or by Klein’s 4-group Kl, we can enlarge the affine Weyl group W (D
to the affine Weyl group of type F
4 or to the extended affine Weyl group of type D
4 ) = S4 ⋉W (D
4 ) ⊃ W̃ (D
4 ) = Kl⋉W (D
Definition 4.6 Replacing the group W (D
4 ) with W (F
4 ) in Definition 3.2, we can define
a coarser stratification of K than the W (D(1)4 )-stratification, called the W (F
4 )-stratification.
Moreover, replacingW (D
4 ) with W̃ (D
4), we can also think of a stratification of K intermediate
between these two stratifications, called the W̃ (D
4 )-stratification.
The following is the classification of the W (F
4 )-strata and W̃ (D
4 )-strata.
Lemma 4.7 For each abstract Dynkin type ∗ in Table 2, there is a unique W (F (1)4 )-stratum of
type ∗. As for the W̃ (D(1)4 )-strata, we have the following classification (see also Figure 9).
(1) For ∗ ∈ {D4, A⊕41 , A⊕31 , A2, A1, ∅}, there is a unique W̃ (D
4 )-stratum of abstract type ∗
and this unique stratum is denoted by the same symbol ∗.
(2) For ∗ ∈ {A3, A⊕21 }, there are exactly three W̃ (D
4 )-strata of abstract type ∗;
(a) for ∗ = A3, the stratum (A3)i represented by I = {0, j, k} with {i, j, k} = {1, 2, 3};
(b) for ∗ = A⊕21 , the stratum (A⊕21 )i represented by I = {j, k} with {i, j, k} = {1, 2, 3}.
(A⊕21 )i
(A3)i
Figure 10: Adjacency relations among W̃ (D
4 )-strata (i = 1, 2, 3)
If something about the transformation g2j is discussed for a fixed index j, the relevant
stratification is the W̃ (D
4 )-stratification. Namely we may discuss the issue on each W̃ (D
stratum, choosing any representative of each W̃ (D
4 )-orbit, since in the commutative diagram
(35) we have τ(j) = j and hence g2
= g2j for every σ ∈ Kl (see also Remark 5.1). For two
W̃ (D
4 )-strata, say ∗ and ∗∗, we write ∗ → ∗∗ if the stratum ∗∗ lies on the boundary of the
stratum ∗. All the possible adjacency relations ∗ → ∗∗ are depicted in Figure 10. Note that
there are no adjacency relations between (A⊕21 )i and (A3)j for any distinct i, j ∈ {1, 2, 3}.
5 Bäcklund Transformations
In this section we briefly discuss Bäcklund transformations, especially the characterization of
them in terms of Riemann-Hilbert correspondence [15, 16]. This topic is included here in order
to confirm that our problem may be treated modulo Bäcklund transformations.
For each σ ∈ S4 we define the isomorphism of affine cubic surfaces
σ : S(θ) → S(σ(θ)), (x1, x2, x3) 7→ (ετ(1)xτ(1), ετ(2)xτ(2), ετ(3)xτ(3)),
where σ ∈ S4 is identified with (τ, ε) ∈ S3 ⋉ Kl via the isomorphism (34). Consider the
natural homomorphism W (F
4 ) = S4 ⋉ W (D
4 ) → S4, w 7→ σ. Since the Riemann-Hilbert
correspondence (12) is an analytic minimal resolution of singularities, for each w ∈ W (F (1)4 ),
there exists an analytic isomorphism w : Mz(κ) → Mz(w(κ)) such that the diagram
Mz(κ)
w−−−→ Mz(w(κ))
RHz,κ
yRHz,w(κ)
S(θ) −−−→
S(σ(θ))
is commutative, for any fixed κ ∈ K with θ = rh(κ) ∈ Θ.
The commutative diagram (36) characterizes the Bäcklund transformations of Painlevé VI.
Namely the map w : Mz(κ) → Mz(w(κ)) turns out to be algebraic and there are suitable affine
coordinates on Mz(κ) and Mz(w(κ)) in terms of which the map w can be represented by the
usual formula for Bäcklund transformations known as birational canonical transforamtions [27]
(see [15, 16] for the precise statement). In other words the Riemann-Hilbert correspondence is
equivariant under the Bäcklund transformations and so is our main problem.
Remark 5.1 The S4-factor of W (F
4 ) = S4 ⋉W (D
4 ) or more strictly the S3-factor of S4 =
S3⋉Kl permutes the three fixed singular points 0, 1 and ∞, while they are fixed by W̃ (D(1)4 ) =
Kl ⋉ W (D
4 ). Hence we may consider our problem only around the origin z = 0 and, upon
restricting our attention to z = 0, we may discuss it modulo the Bäcklund action of W̃ (D
6 Fixed Points and Periodic Points
We shall more closely investigate the fixed point set Fixj(θ), or rather its subset Fix
j (θ)
j (θ) of smooth fixed points, by solving the system of equations (23). In view of (21) the last
two equations in (23) are expressed as a linear system for the unknowns (xj , xk),
2xj + xixk = θj ,
xixj + 2xk = θk,
If its determinant 4− x2i is nonzero, then system (37) is uniquely settled as
2θj − xiθk
4− x2i
, xk =
2θk − xiθj
4− x2i
. (38)
Substituting (38) into equation f(x, θ) = 0 yields a quartic equation for the unknown xi,
x4i − θix3i + (θ4 − 4)x2i + (4θi − θjθk)xi + θ2j + θ2k − 4θ4 = 0. (39)
Conversely, if xi is a root of equation (39) with nonzero x
i − 4, then subsituting this into
formula (38) yields a root of system (23). The four roots of quartic equation (39) are given by
F (bi, b4; bj , bk), F (bi, b
4 ; bj, bk), F (bj, bk; bi, b4), F (bj , b
k ; bi, b4),
counted with multiplicities, where F (bi, b4; bj , bk) is defined by
F (bi, b4; bj , bk) = bib4 + b
4 . (40)
We pick up the root xi = F (bi, b4; bj, bk). Note that F (bi, b4; bj, bk)
2 − 4 is nonzero precisely
when b2i b
4 6= 1. If this is the case, then substituting xi = F (bi, b4; bj , bk) into formula (38) yields
xj = G(bi, b4; bj , bk) and xk = G(bi, b4; bk, bj), where G(bi, b4; bj , bk) is defined by
G(bi, b4; bj , bk) =
(bi + b4)(bj + bk)(bjbk + 1)
2(bib4 + 1)bjbk
(bi − b4)(bj − bk)(bjbk − 1)
2(bib4 − 1)bjbk
. (41)
Therefore, if P (bi, b4; bj, bk) denotes the point defined by
xi = F (bi, b4; bj, bk), xj = G(bi, b4; bj , bk), xk = G(bi, b4; bk, bj),
then x = P (bi, b4; bj , bk) gives a root of system (23) with nonzero x
i −4 provided that b2i b24 6= 1.
If x is at this root, then yi(x, θ) admits the following nice factorization
yi(x, θ) =
(bi − b−1i )(b4 − b−14 )
(b2i b
4 − 1)2
(εj ,εk)∈{±1}
k b4 − 1)
= (bib4 − b−1i b−14 )−2 {F (bi, b4; bj, bk)− F (bi, b−14 ; bj, bk)}
{F (bi, b4; bj, bk)− F (bj , bk ; bi, b4)}
{F (bi, b4; bj, bk)− F (bj , b−1k ; bi, b4)}.
label fixed point existence smoothness condition
1 P (bi, b4; bj , bk) κi + κ4 6∈ Z κi 6∈ Z, κ4 6∈ Z, κi + κ4 ± κj ± κk 6∈ 2Z+ 1
2 P (bi, b
4 ; bj, bk) κi − κ4 6∈ Z κi 6∈ Z, κ4 6∈ Z, κi − κ4 ± κj ± κk 6∈ 2Z+ 1
3 P (bj, bk; bi, b4) κj + κk 6∈ Z κj 6∈ Z, κk 6∈ Z, κj + κk ± κi ± κ4 6∈ 2Z+ 1
4 P (bj, b
k ; bi, b4) κj − κk 6∈ Z κj 6∈ Z, κk 6∈ Z, κj − κk ± κi ± κ4 6∈ 2Z+ 1
Table 3: Smooth fixed points x ∈ Fix◦j(θ) with nonzero x2i − 4
Hence P (bi, b4; bj, bk) is a smooth point of S(θ) if and only if F (bi, b4; bj , bk) is a simple root of
equation (39). In terms of κ ∈ K, the existence and smoothness conditions for P (bi, b4; bj , bk)
are given by κi + κ4 6∈ Z and κi 6∈ Z, κ4 6∈ Z, κi + κ4 ± κj ± κk 6∈ 2Z+ 1, respectively.
Lemma 6.1 The smooth fixed points x ∈ Fix◦j (θ) with nonzero x2i −4 are precisely those points
in Table 3 which satisfy the existence and smoothness conditions mentioned there.
The fixed points in Table 3 is closely related to the configuration of lines on the affine cubic
surface S(θ) or on its compactification S(θ) by the standard embedding
S(θ) →֒ S(θ) ⊂ P3, x = (x1, x2, x3) 7→ [1 : x1 : x2 : x3],
where the projective cubic surface S(θ) is defined by the homogeneous equation
F (X, θ) := X1X2X3 +X0(X
3 )−X20 (θ1X1 + θ2X2 + θ3X3) + θ4X30 = 0.
It is obtained from the affine surface S(θ) by adding three lines at infinity
Li = {X ∈ P3 : X0 = Xi = 0 } (i = 1, 2, 3),
whose union L = L1 ∪ L2 ∪ L3 is called the tritangent lines at infinity.
It is well known that a smooth projective cubic surface has exactly 27 lines on it. We
describe them in the current situation [20]. Let Li(bi, b4; bj, bk) be the line in P
3 defined by
Xi = (bib4 + b
4 )X0, Xj + (bib4)Xk = {bi(bk + b−1k )}+ b4(bj + b−1j )X0. (43)
1 L+i1 = Li(bi, b4; bj, bk) L
i1 = Li(b
i , b
4 ; bj, bk)
2 L+i2 = Li(bi, b
4 ; bj, bk) L
i2 = Li(b
i , b4; bj, bk)
3 L+i3 = Li(bj , bk; bi, b4) L
i3 = Li(b
j , b
k ; bi, b4)
4 L+i4 = Li(bj , b
k ; bi, b4) L
i4 = Li(b
j , bk; bi, b4)
Table 4: Eight lines intersecting the line Li at infinity, divided into four pairs
L−24L
L+11 L
L−12 L
Figure 11: The 27 lines on a smooth cubic surface viewed from the tritangent lines at infinity
For each i ∈ {1, 2, 3} the eight lines in Table 4 are the only lines on S(θ) that intersect the i-th
line Li at infinity, but they do not intersect the remaining two lines Lj and Lk at infinity. These
lines are divided into four pairs as in Table 4. The surface S(θ) is always smooth at infinity
[20] and hence, if κ ∈ K−Wall, then S(θ) is smooth everywhere. In this case, the two lines in
the same pair intersect, while two lines from different pairs do not. The intersection point of
the i-th pair is exactly the i-th fixed point in Table 3. See Figure 11 for a total image of these
situations. Caution: for a pair of distinct indices i and j, the intersection relations between L±iµ
and L±jν are not depicted in the Figure 11. We also remark that in some degenerate cases the
lines L+iµ and L
iµ may meet in a point on the line Li at infinity.
Next we consider the case where the determinant 4 − x2i of system (37) vanishes. In other
words we ask when the fixed point set Fixj(θ) contains points x such that xi ∈ {±2}.
Lemma 6.2 Fixj(θ) contains a point x such that xi = 2δ with δ ∈ {±1} if and only if either
(1) bib4 = bib
4 = δ; or
(2) bjbk = bjb
k = δ; or
(3) bib
4 = bjb
k = δ for some double sign (εk, ε4) ∈ {±1}2.
If this is the case, then θk = δθj and all such poins x are exactly those points on the line
ℓδj := { xi = 2δ, xj + δxk = θj/2 }. (44)
In particular ℓδj ⊂ Fixj(θ) precisely when xi = 2δ is a multiple root of the quartic equation (39).
xi ∈ {±2} multiplicity component remark
no simple smooth point intersection point of L±iµ
no multiple singular point Riccati locus
yes multiple line ℓ+j or ℓ
j line contains singular points
yes simple empty L±iµ intersects at infinity
Table 5: The roots of quartic equation (39) and the components of Fixj(θ)
Proof. If xi = 2δ with δ ∈ {±1} then system (37) is linearly dependent, so that θk − δθj = 0.
However, since θk − δθj = (bibj)−1(bib4 − δ)(bib−14 − δ)(bjbk − δ)(bjb−1k − δ), we have either
4 = δ for some sign ε4 ∈ {±1} or bjb
k = δ for some sign εk ∈ {±1}. Taking the equation
xj + δxk = θj/2 into account, we observe that f(x, θ) factors as
f(x, θ) =
−(2bibj)−2(bib−ε44 − δ)2(bjbk − δ)2(bjb−1k − δ)2 (if bib
4 = δ),
−(2bibj)−2(bjb−εkk − δ)2(bib4 − δ)2(bib−14 − δ)2 (if bjb
k = δ).
If bib
4 = δ then equation f(x, θ) = 0 yields either bib
4 = δ or bjb
k = δ for some sign
εk ∈ {±1}; the former case falls into case (1) while the latter falls into case (3). In a similar
manner the other case bjb
k = δ falls into case (2) or case (3).
Next, if Fixj(θ) contains the line ℓ
j , then what we have just proved implies that
F (bi, b4; bj, bk) = F (bi, b
4 ; bj, bk) = 2δ if condition (1) is satisfied;
F (bj, bk; bi, b4) = F (bj , b
k ; bi, b4) = 2δ if condition (2) is satisfied;
F (bi, b
4 ; bj, bk) = F (bj , b
k ; bi, b4) = 2δ if condition (3) is satisfied.
Hence xi = 2δ is a multiple root of the quartic equation (39). Conversely, if xi = 2δ is a
multiple root of (39), then we can trace the argument backwards to conclude that the system
(23) admits the line solution ℓδj , that is, Fixj(θ) contains ℓ
j . ✷
Summarizing the arguments so far yields a classification of the irreducible components of
the algebraic set Fixj(θ) in terms of certain roots of quartic equation (39).
Theorem 6.3 Any irreducible component of Fixj(θ) is just a single point or a single affine line;
the former is called a point component and the latter is called a line component respectively. The
irreducible components of Fixj(θ) are in one-to-one correspondence with those roots of quartic
equation (39) which are not a simple root x = (x1, x2, x3) such that xi ∈ {±2}.
(1) A simple root with xi 6∈ {±2} corresponds to a point component that is a smooth point of
the surface S(θ) and is given in Table 3.
(2) A multiple root with xi 6∈ {±2} corresponds to a point component that is a singular point
of the surface S(θ) and is associated with Riccati solutions.
(3) A multiple root with xi ∈ {±2} corresponds to a line component; either ℓ+j or ℓ−j .
(4) A simple root with xi ∈ {±2} corresponds to no component of Fixj(θ).
A summary of Theorem 6.3 is given in Table 5 and the following remark may be helpful.
Remark 6.4 The assertions (3) and (4) of Theorem 6.3 may be well understood through the
degeneration of line configration on the projective surface S(θ) as the parameter θ = rh(κ)
tends to a special position. For a generic value of θ the lines L±iµ intersect in a single (smooth)
point on the affine part S(θ) of S(θ). If the parameter θ tends to a special position so that
a corresponding root xi of quartic equation (39) approaches {±2}, then the two line L±iµ are
getting “parallel” and eventually either coincide completely or meet in a point at infinity. The
former case falls into assertion (3) and the latter case falls into assertion (4) respectively.
Let us investigate more closely the case where Fixj(θ) contains line components.
Lemma 6.5 Let θ = rh(κ) with κ ∈ K and (i, j, k) be any cyclic permutation of (1, 2, 3).
(1) Fixj(θ) contains either ℓ
j or ℓ
j but not both of them if and only if κ lies in a W̃ (D
stratum appearing in the following adjacency diagram (see also Figure 10) :
(A⊕21 )i −−−→ A⊕31y
(A3)i −−−→ D4
(2) Fixj(θ) contains both ℓ
j and ℓ
j if and only if θ = (0, 0, 0,−4), that is, precisely when κ
is in the W̃ (D
4 )-stratum of type A
1 . In this case one has Fixj(θ) = ℓ
j ∐ ℓ−j .
Proof. Lemma 6.2 implies that Fixj(θ) contains at least one of ℓ
j and ℓ
j if and only if either (a)
bib4 = bib
4 ∈ {±1}; or (b) bjbk = bjb−1k ∈ {±1}; or (c) bib
4 = bjb
k ∈ {±1} for some double
sign (εk, ε4) ∈ {±1}2. This property is invariant under the action of W̃ (D(1)4 ) = Kl⋉W (D
on K. Using this action we can reduce conditions (a) and (c) to condition (b). First, observe
that the permutation (i, j)(k4) ∈ Kl induces the map (b0, bi, bj , bk, b4) 7→ (b0, bj , bi,−b4,−bk),
which reduces condition (a) to condition (b). Next, formula (13) implies that the reflection wi
induces the multiplicative transformation wi : B → B, b 7→ b′, where
b′j =
−bjbciji (i = 4, j = 0),
i (otherwise).
Applying w4 or wk if necessary, we may assume from the beginning that ε4 = 1 and εk = −1
in condition (c). Then using w0 there yields b
0bib4 = bjb
k ∈ {±1}. But since b20bib4bjbk = 1,
we have bjbk = bjb
k ∈ {±1}, that is, condition (b). Note that condition (b) means κj, κk ∈ Z.
On the other hand, the extended affine Weyl group W̃ (D
4 ) contains shifts
(κ0, κi, κj, κk, κ4) 7→ (κ0, κi − 1, κj + 1, κk, κ4),
(κ0, κi, κj, κk, κ4) 7→ (κ0, κi, κj, κk + 1, κ4 − 1).
Repeated applications of these operations and their inverses can shift κj and κk independently
by arbitrary integers. Thus the condition κj, κk ∈ Z can further be reduced to κj = κk = 0.
Thus we have shown that if Fixj(θ) contains at least one of ℓ
j and ℓ
j , then κ must lie in
the W̃ (D
4 )-stratum of type (A
1 )i or on its boundary strata of types (A3)i, A
1 , D4, A
Moreover, it is easy to see that the converse is also true.
For a sign δ ∈ {±1} the conditions (1), (2), (3) in Lemma 6.2 are denoted by (1δ), (2δ), (3δ),
respectively. Now we assume that Fixj(θ) contains both ℓ
j and ℓ
j . Then there exists a pair
of conditions, one from {(1+), (2+), (3+)} and the other from {(1−), (2−), (3−)}, that are valid
at the same time. Such a pair can be consistent only if it is either (1−) + (2−); or (2+) + (1−);
or (3+) + (3−) where if the sign for (3+) is (εk, ε4) then the sign for (3
−) must be its antipode
(−εk,−ε4). The first and second pairs lead to b21 = b22 = b23 = b24 = 1 and to b1b2b3b4 = −1,
while the third pair yields b21 = b
2 = b
3 = b
4 = −1. These are nothing but the conditions (a)
and (b) in Example 3.4. (1). Therefore κ must lie in the stratum of type A⊕41 . Combining this
with the discussion in the last paragraph establishes the assertion (1), as well as a large part
of the assertion (2). The only thing yet to be proved is the assertion that if Fixj(θ) contains
both ℓ+j and ℓ
j , then Fixj(θ) = ℓ
j ∐ ℓ−j . For this, the last part of Lemma 6.2 implies that both
xi = 2 and xi = −2 are multiple roots of the quartic equation (39), so that there are no other
roots of the equation (39). Thus Fixj(θ) has no elements other than those in ℓ
j ∐ ℓ−j . ✷
Now we turn our attention to periodic points and investigate the set P̃er
j(θ;n) of periodic
points of prime period n > 1 on the non-Riccati locus.
Lemma 6.6 For any integer n > 1 the set P̃er
j(θ;n) is biholomorphic to the disjoint union of
ϕ(n) copies of C×, where ϕ(n) denotes the number of integers 0 < m < n coprime to n.
Proof. By Lemma 4.5 we can identify P̃er
j(θ;n) with Perj(θ;n) and hence may work downstairs.
For any integer 0 < m < n coprime to n, we consider the projective curve Cm in P
3 defined by
{4 cos2(πm/n)− 2θi cos(πm/n) + θ4}X20 −X0(θjXj + θkXk)
+X2j +X
k + 2 cos(πm/n)XjXk = 0, (47)
Xi − 2 cos(πm/n)X0 = 0, (48)
where (47) is obtained from F (X, θ) = 0 by substituting (48) and factoring X0 out of it. It
follows from −2 < 2 cos(πm/n) < 2 that Cm is an irreducible smooth conic curve. By equations
(24) of Lemma 4.4 the closure Perj(θ;n) of Perj(θ;n) in S(θ) is the union of these ϕ(n) curves
Cm. The curve Cm intersects the lines L = Li ∪ Lj ∪ Lk at infinity in the two points
P±m : [X0 : Xi : Xj : Xk] = [0 : 0 : −1 : exp(±π
−1m/n)] ∈ Li.
If Cm := Cm−{P+m , P−m}, then Cm is biholomorphic to C×, since Cm ∼= P1. So Perj(θ;n) is the
disjoint union of the ϕ(n) curves Cm with 0 < m < n, (m,n) = 1, and hence biholomorphic to
the disjoint union of ϕ(n) copies of C×. ✷
7 Case-by-Case Study
We make case-by-case studies of F̃ixj(θ) and P̃er
j(θ;n) according to the adjacency diagram in
Figure 10. Now we need to introduce some notation. Recall that we have the resolution of
singularities (15) which restricts to an isomorphism ϕ : S̃◦(θ) → S◦(θ) and that the smooth
fixed points Fix◦j(θ) in S◦(θ) are listed in Table 3. For each P ∈ Fix◦j(θ) let P̃ ∈ F̃ix
j (θ)
denote its lift through the isomorphism ϕ. For example, P̃ (bi, b4; bj , bk) denotes the lift of
P (bi, b4; bj, bk). If {· · · } is a set of expressions P̃ (bi, b±14 ; bj, bk), P̃ (bj , b±1k ; bi, b4), then we denote
by {{· · · }} its subset obtained by discarding those expressions which do not satisfy either the
existence condition or the smoothness condition of Table 3. An example is given in (49) below.
Example 7.1 (∅) Consider the W̃ (D
4 )-stratum of type ∅, namely, the big open K −Wall.
F̃ixj(θ) = {{ P̃ (bi, b4; bj , bk), P̃ (bi, b−14 ; bj, bk), P̃ (bj , bk; bi, b4), P̃ (bj , b−1k ; bi, b4) }}. (49)
Here we have only to care the existence condition, as we are in the big open where the smooth-
ness condition is fulfilled by hypothesis. If a finer stratification of K attached to the W (F (1)4 )-
action on K is introduced, then a more precise description of (49) is feasible, detecting how
many and which elements are there in (49), but the details are omitted. We only remark that
F̃ixj(θ) consists of four distinct points in the most generic case where none of κi±κ4 and κj±κk
are integers. As for the periodic points, since there is no Riccati locus, we have
P̃erj(θ;n) = P̃er
j (θ;n), P̃er
j(θ;n) = ∅, (n > 1).
Example 7.2 (A1) Consider the W̃ (D
4 )-stratum of type A1. We may assume that κ0 = 0 so
that b0 = 1 and bibjbkb4 = 1. Note that none of b
i , b
j , b
4 equals 1. We claim that b
k 6= 1.
Otherwise, we would have κj + κk ∈ Z. Applying a shift as in (46) to κ repeatedly, we may
assume that κj + κk = 0 while keeping the condition κ0 = 0. Then the transformation w0wj
sends κ to κ′ with κ′j = 0 and κ
k = κj + κk = 0, so that one has κ ∈ K{j,k}, namely, κ lies in
the closure of the stratum of type (A⊕21 )i. This contradicts the assumption that we are in the
stratum of type A1. In this case, the surface S(θ) has a unique singular point of type A1 at
(xi, xj , xk) = (bib4 + b
4 , bjb4 + b
4 , bkb4 + b
Blow up S(θ) at this point to obtain a minimal resolution (15). Write the blowing-up as
(xi, xj, xk) = (uiuj + bib4 + b
4 , uj + bjb4 + b
4 , ukuj + bkb4 + b
in terms of coordinates (ui, uj, uk). The exceptional set e is the irreducible quadratic curve
uj = bibjbk + (1 + b
j )bkui + (1 + b
j )biuk + bibjbk(u
i + u
k) + (1 + b
k)bjuiuk = 0,
which can be paramatrized as uj = 0 and
b2i b
k − 1)3t
bj{bk(b2i − 1)(b2j − 1) + bi(b2jb2k − 1)t}{(b2k − 1)(b24 − 1) + bibkb24(b2jb2k − 1)t}
{b2jbk(b2i − 1)(b2k − 1)− bi(b2jb2k − 1)t}{bk(b2j − 1)(1− b24) + bib24(b2jb2k − 1)t}
bj{bk(b2i − 1)(b2j − 1) + bi(b2jb2k − 1)t}{(b2k − 1)(b24 − 1) + bibkb24(b2jb2k − 1)t}
In terms of this parametrization, the lifted transformation g̃2j acts on the exceptional curve
e ≃ P1 by the multiplication t 7→ b2jb2kt. Since b2jb2k 6= 1, the set F̃ix
j(θ) consists of the two
p0 p+
S̃(θ)
t = 0 t = ∞ s = t = 0 s = ∞
t = ∞
S̃(θ)
Figure 12: Surface of types A1 (left) and A2 (right)
points, say p and q, corresponding to t = 0 and t = ∞ (see Figure 12, left). On the other hand,
the possible candidates for the smooth fixed points F̃ix
j (θ) are only the points of labels 2 and
4 in Table 3, since those of labels 1 and 3 do not satisfy the smoothness condition. Thus,
F̃ixj(θ) = {{P̃ (bi, b−14 ; bj , bk), P̃ (bj , b−1k ; bi, b4)}} ∐ {p, q}. (50)
As for the Riccati periodic points P̃er
j(θ;n), the discussion above implies that for any n > 1,
j(θ;n) =
e (if bjbk is a primitive 2n-th root of unity),
∅ (otherwise).
Example 7.3 (A2) Consider the W̃ (D
4 )-stratum of type A2. We may assume that κ0 =
κi = 0 so that b0 = bi = 1. Then the surface S(θ) has a unique singular point of type A2 at
(xi, xj , xk) = (b4 + b
4 , bjb4 + b
4 , bkb4 + b
Blow up S(θ) at this point to obtain a minimal resolution (15). Write the blowing-up as
(xi, xj , xk) = (uiuj + b4 + b
4 , uj + bjb4 + b
4 , ukuj + bkb4 + b
in terms of coordinates (ui, uj, uk). The exceptional set e is the union of two lines
e+ : uj = bkui + uk + bjbk = 0, e
− : uj = b
k ui + uk + b
k = 0,
intersecting in a point. These lines are parametrized as
e+ : (ui, uj, uk) =
k − 1
bj(1− b2k) + (b2jb2k − 1)s
bk(1− b2j ) + bjbk(1− b2jb2k)s
bj(1− b2k) + (b2jb2k − 1)s
e− : (ui, uj, uk) =
k − 1
bj(1− b2k) + (b2jb2k − 1)t
bk(1− b2j ) + b−1j b−1k (1− b2jb2k)t
bj(1− b2k) + (b2jb2k − 1)t
with the intersection point corresponding to s = t = 0. In terms of these parametrizations, the
lifted transformation g̃2j acts on e
+ and e− by the multiplications s 7→ b−2j b−2k s and t 7→ b2jb2kt,
ℓ̃+j C
ei e4
S̃(θ)
Figure 13: Surface of type (A⊕21 )i
which are rewritten as s 7→ b24s and t 7→ b−24 t, since bjbkb4 = 1. Note that b24 6= 1, for otherwise
κ would be in the closure of the W̃ (D
4 )-stratum of type (A3)i. So g̃
j has exactly two fixed
points p0 and p+ on e
+ corresponding to s = 0 and s = ∞. Similarly g̃2j has exactly two fixed
points p0 and p− on e
− corresponding to t = 0 and t = ∞, where p0 is the intersection point of
e+ and e− (see Figure 12, right). Thus we have F̃ix
j(θ) = {p0, p+, p−}. Next we consider the
smooth fixed point of g̃2j on S̃(θ). Since we are assuming that κi = κj + κk + κ4 = 1, the points
of labels 1, 2, 3 in Table 3 do not satisfy the smoothness condition and that of label 4 is the
only smooth fixed point. Thus F̃ix
j (θ) = {P̃ (bj , b−1k ; bi, b4)} and hence
F̃ixj(θ) = {P̃ (bj , b−1k ; bi, b4), p0, p+, p−}. (51)
In the remaining cases presented below, Fixj(θ) contains at least one line component.
Example 7.4 (A
) First we consider F̃ixj(θ) and P̃er
j(θ) on the W̃ (D
4 )-stratum of type
(A⊕21 )i. We may assume that κj = κk = 0 so that bj = bk = 1. Since our stratum is not of type
(A3)i nor of type D4, we have (bib4−1)(bib−14 −1) 6= 0 or equivalently bi+b−1i 6= b4+b−14 . In this
case Fixj(θ) contains the line ℓ
j but does not the line ℓ
j and the surface S(θ) has two singular
points of type A1 at (xi, xj , xk) = (2, bi + b
i , b4 + b
4 ) and (xi, xj , xk) = (2, b4 + b
4 , bi + b
We denote the former singularity by qi and the latter by q4 respectively; both singularities lie on
the line ℓ+j . Blow up S(θ) at these points to obtain a minimal resolution as in (15). Let ℓ̃+j be
the strict transform of ℓ+j , and let ei and e4 be the exceptional curves over qi and q4 respectively.
Moreover let pi be the intersection point of ℓ̃
j and ei. Similarly let p4 be the intersection point
of ℓ̃+j and e4 (see Figure 13). Then the blowing-up at the point qi is represented as
(xi, xj, xk) = (uiuj + 2, uj + bi + b
i , ukuj + b4 + b
in terms of coordinates (ui, uj, uk) around (0, 0, 0). The strict transform ℓ̃
j and the exceptional
curve ei are given by ui = uk + 1 = 0 and
uj = (bib4)(u
i + u
k) + (b
i + 1)b4(uiuk) + bi(b
4 + 1)ui + 2(bib4)uk + (bib4) = 0.
The exceptional curve ei admits a parametrization
(bib4 − 1)(bib−14 − 1)
(t+ bi)(bit + 1)
, uj = 0, uk = −
bi(t+ b4)(b4t+ 1)
b4(t+ bi)(bit+ 1)
, (52)
where the intersection point pi has coordinates (ui, uj, uk) = (0, 0,−1), which corresponds to
t = ∞. The lifted transformation g̃2j acts on ei as a Möbius transformation fixing pi. Some
computations show that in terms of the variable t this transformation is just the shift
t 7→ t+ (bi + b−1i )− (b4 + b−14 ).
and hence a parabolic transformation. Thus g̃2j has no periodic points on ei other than the
fixed point pi. By symmetry, g̃
j also acts on e4 as a parabolic Möbius transformation fixing p4
only. Summarizing the arguments, we conclude that on the W̃ (D
4 )-stratum of type (A
1 )i,
F̃ixj(θ) = ℓ̃
j ∐ { P̃ (bi, b4; bj , bk), P̃ (bi, b−14 ; bj, bk) }, P̃er
j(θ;n) = ∅ (n > 1). (53)
Next we consider F̃ixi(θ) on the W̃ (D
4 )-stratum of type (A
1 )i. Some calculations show
that there are parametrizations of ei and e4 such that g̃
i acts on ei and e4 as the multiplications
t 7→ b24t and t 7→ b2i t respectively. (Modify (52) to get such parametrization.) Since b24 6= 1 and
b2i 6= 1, the transformation g̃2i has exactly two fixed points, say pii and qii, on ei, and exactly
two fixed points, say pi4 and qi4, on e4. There are no smooth fixed points F̃ix
i (θ), because the
smoothness condition of Table 3 with (i, j, k) replaced by (k, i, j) is not satisfied for any labels
there. Thus we have F̃ixi(θ) = F̃ix
i (θ) = {pii, qii, pi4, qi4} and F̃ix
i (θ) = ∅. By symmetry
there is a similar characterization of F̃ixk(θ). By permuting the indices (i, j, k), we have
F̃ixj(θ) = F̃ix
j(θ) = {four points}, F̃ix
j(θ) = ∅, (54)
on the W̃ (D
4 )-strata of types (A
1 )j and (A
1 )k. A slightly further consideration yields
j(θ;n) =


ej ∐ e4 (if bj and b4 are primitive 2n-th roots of unity),
ej (if b4 is a primitive 2n-th root of unity, but bj is not),
e4 (if bj is a primitive 2n-th root of unity, but b4 is not),
∅ (otherwise).
on the stratum (A⊕21 )j and a similar characterization of it on the stratum (A
1 )k.
Example 7.5 (A3) First we consider F̃ixj(θ) and P̃er
j(θ) on the W̃ (D
4 )-stratum of type
(A3)i. We may assume that κ0 = κj = κk = 0 and κi+κ4 = 1 so that bj = bk = 1 and bib4 = 1.
But we have bib
4 6∈ {±1}, since our stratum is not of type D4. In this case Fixj(θ) contains
the line ℓ+j but does not the line ℓ
j . The surface S(θ) has only one singular point of type A3
at (xi, xj , xk) = (2, b4 + b
4 , b4 + b
4 ), which lies on the line ℓ
j . Blow up the singular point.
This blowing-up is expressed as (xi, xj , xk) = (uiuj + 2, uj + b4 + b
4 , ukuj + b4 + b
4 ) in terms
of coordinates (ui, uj, uk) around (0, 0, 0). The strict transform of the surface S(θ) is given by
uj = b4uiujuk + b4u
i ++b4u
k + (b
4 + 1)uiuk + (b
4 + 1)ui + 2b4uk + b4 = 0,
which has yet one singular point, say q. The exceptional curve consists of two line components
uj = ui+b4uk+b4 = 0 and uj = b4ui+uk+1 = 0, whose intersection point (ui, uj, uk) = (0, 0,−1)
is exactly the singular point q. The strict transform of ℓ+j is now given by ui = uk+1 = 0, which
ℓ̃+jej ek
pj p0 pk
S̃(θ)
Figure 14: Surface of type (A3)i
also passes through q. Blow up again the singular point q. Let e0 be the exceptional curve and
let ej, ek, ℓ̃
j be the strict transforms of the lines uj = ui+ b4uk+ b4 = 0, uj = b4ui+uk+1 = 0,
ui = uk + 1 = 0 respectively. If we express this blowing-up as (ui, uj, uk) = (vi, vivj , vivk − 1),
then the exceptional curve e0 is given by vi = b4 − b4vj + (b24 + 1)vk + b4v2k = 0; ej is given by
vj = 1 + b4vk = 0; and ek is given by vj = b4 + vk = 0. The intersection point of e0 and ej is
(vi, vj , vk) = (0, 0,−bi) and that of e0 and ek is (vi, vj , vk) = (0, 0,−b4). If ej is parametrized as
(vi, vj , vk) = ((t+bi)
−1, 0,−bi), then the transformation g̃2j acts on ej as the shift t 7→ t+b4−bi.
Similarly, if ek is parametrized as (vi, vj, vk) = ((t + b4)
−1, 0,−b4), then g̃2j acts on ek as the
shift t 7→ t + b4 − bi. Hence g̃2j acts on ej and ek as parabolic Möbius transformations fixing
only pj and qj . Then g̃
j acts on e0 as the identity, because it also fixes the intersection point
p0 of e0 and ℓ̃
j . Summarizing the arguments we see that on the stratum of type (A3)i,
F̃ixj(θ) = ℓ̃
e0 ∐ { P̃ (bi, b−14 ; bj , bk) }, P̃er
j(θ;n) = ∅ (n > 1), (55)
where ℓ̃+j ∪
e0 indicates that the curves ℓ̃
j and e0 meet in the point p0.
Next we consider F̃ixi(θ) and P̃er
i (θ;n) on the W̃ (D
4 )-stratum of type (A3)i. If we take a
parametrization of e0 such that t = 0 and t = ∞ correspond to the points pj and pk respectively,
then a simple check shows that the transformation g̃2i on ej is expressed as t 7→ b−24 t. There
is a parametrization of ej such that t = 0 corresponds to pj and g̃
i is given by t 7→ b24t. Since
b24 6= 1, the transformation g̃2i has exactly two fixed points on ej , one of which is just pj and the
other is denoted by pij . Similarly, there is a parametrization of ek such that t = 0 corresponds
to pk and g̃
i is given by t 7→ b−24 t, and hence g̃2i has exactly two fixed points on ek, one of
which is just pk and the other is denoted by pik. There are no smooth fixed points F̃ix
i (θ),
because the smoothness condition of Table 3 with (i, j, k) replaced by (k, i, j) is not satisfied
for any labels there. So we have F̃ixi(θ) = F̃ix
i (θ) = {pj, pij , pk, pik} and F̃ix
i (θ) = ∅ on the
W̃ (D
4 )-stratum of type (A3)i. By symmetry there is a similar characterization of F̃ixk(θ) on
the same stratum. By permuting the indices (i, j, k), we have
F̃ixj(θ) = F̃ix
j(θ) = {four points}, F̃ix
j(θ) = ∅, (56)
ej ek
pj pk
r0 r∞
S̃(θ)
Figure 15: Surface of type A⊕31
on the W̃ (D
4 )-strata of types (A3)j and (A3)k. A slightly further consideration yields
j(θ;n) =
ei (if b4 is a primitive 2n-th root of unity),
∅ (otherwise),
on the stratum (A3)j and a similar characterization of it on the stratum (A3)k.
Example 7.6 (A
) Consider the W̃ (D
4 )-stratum of type A
1 . We may assume that κi =
κj = κk = 0 so that bi = bj = bk = 1. But we have b4 6∈ {±1} since our stratum is not of
type D4 nor of type A
1 . In this case the surface S(θ) has three singular points of type A1 at
(xi, xj, xk) = (b4+b
4 , 2, 2), (2, b4+b
4 , 2), (2, 2, b4+b
4 ), which are called qi, qj , qk respectively.
Note that the two points qj and qk lie on the line ℓ
j but qi does not lie on the union ℓ
j ∐ ℓ−j .
The minimal resolution (15) is obtained by blowing up these three points (see Figure 15). First,
consider the blowing-up at qk and represent it by (xi, xj , xk) = (uiuj+2, uj+2, ukuj+b4+b
Then the strict transform ℓ̃+j of the line ℓ
j is given by ui = uk + 1 = 0, while the exceptional
curve ek is given by b4(ui + uk + 1)
2 + (b4 − 1)2ui = 0. The curves ℓ̃+j and ek intersect in the
point (ui, uj, uk) = (0, 0,−1); this point is called pk. If we parametrize the curve ek as
ui = −
(b4 − 1)2t2
, uj = 0, uk = −
{(b4 − 1)t+ 1}{(b4 − 1)t− b4}
(b4 − 1)2t2
(t ∈ P1),
where t = ∞ corresponds to the point pk, then the lifted transformation g̃2j induces the shift
t 7→ t+1 and hence acts on ek as a parabolic Möbius transformation fixing pk only. In a similar
manner g̃2j acts on the exceptional curve ej over qj as a parabolic Möbius transformation fixing
only the intersection point pj of ℓ̃
j and ej . Next we consider the blowing-up at qi and represent
it by (xi, xj , xk) = (uiuj + b4 + b
4 , uj + 2, ukuj + 2). Then the exceptional curve ei is given by
b4(ui + uk + 1)
2 + (b4 − 1)2uk = 0, which can be parametrized as
ui = −
(b4 + 1)
(b4t + 1)2
, uj = 0, uk = −
b4(t− 1)2
(b4t+ 1)2
(t ∈ P1).
ℓ̃+kℓ̃
ejei ek
qi qj qk
S̃(θ)
Figure 16: Surface of type D4
In terms of this parametrization, the transformation g̃2j restricts to the map t 7→ b24t on the
exceptional curve ei. Let r0 and r∞ be the points on ei corresponding to t = 0 and t = ∞ respec-
tively. Since b24 6= 1, the map g̃2j acts on ei as a Möbius transformation with exactly two fixed
points r0 and r∞. Hence the set F̃ixj(θ) contains the line component ℓ̃
j and the Riccati compo-
nent {r0, r∞}, but has no smooth-point component, since F (bi, b4; bj , bk) = F (bi, b−14 ; bj, bk) =
b4 + b
4 6∈ {±2} is a double root of the quartic equation (39) (see Theorem 6.3). Thus we have
F̃ixj(θ) = ℓ̃
j ∐ {r0, r∞}. (57)
As for the Riccati periodic points, since g̃2j acts on ei as t 7→ b24t, we have for any n > 0,
j(θ;n) =
ei (if b4 is a primitive 2n-th root of unity),
∅ (otherwise).
Example 7.7 (D4) Consider the W̃ (D
4 )-stratum of type D4, say, the W (D
4 )-stratum with
value θ = (8, 8, 8, 28). In this case the surface S(θ) has only one singular point of type D4 at
(xi, xj, xk) = (2, 2, 2). The minimal resolution (15) is obtained by successive blowing-ups: Blow
up the singular point. If we express the blowing-up as (xi, xj , xk) = (uiuj + 2, uj + 2, ukuj + 2)
in terms of coordinates (ui, uj, uk), then the strict transform of the surface S(θ) is represented
as uiujuk + (ui + uk + 1)
2 = 0. The exceptional curve e is given by uj = ui + uk + 1 = 0. The
strict transforms of ℓ+i and ℓ
j are given by ui+1 = uk = 0 and ui = uk+1 = 0, while the strict
transform of ℓ+k is at infinity and not expressible in terms of the coordinates (ui, uj, uk). The
blow-up surface has three singularities, all of which are of type A1 and located at the points
in which the exceptional curve e intersects the strict transforms of ℓ+i , ℓ
j , ℓ
k . The lifts of the
transformations g2i , g
j , g
k fix the curve e pointwise, since they fix the three singular points on
it. Again blow up these points. Then we obtain a minimal resolution (15) of the surface S(θ)
as depicted in Figure 16, where ei, ej , ek are the exceptional curves over the singular points and
e++−e+−+
e−−− e−++
S̃(θ) P
Figure 17: Surface of type A⊕41
e0, ℓ̃
i , ℓ̃
j , ℓ̃
k are the strict transforms of e, ℓ
i , ℓ
j , ℓ
k , respectively. Being the strict transform
of e, the exceptional curve e0 is fixed pointwise by the lifts g̃
i , g̃
j , g̃
k of g
i , g
j , g
k, and hence
carries rational solutions. Moreover the lift g̃2j fixes ej pointwise. This can be seen without
computation. Since g̃2j is area-preserving and fixes ℓ̃
j pointwise, it has derivative 1 at pj along
the curve ej . So the Möbius transformation on ej induced by g̃
j is either identity or a map
of parabolic type. But the latter is impossible because it has at least two fixed points at pj
and qj (see Figure 16). Hence g̃
j acts on ej as the identity. Next we shall observe that g̃
acts on ei as a parabolic Möbius transformation fixing qi only. If we express the blowing-up
at (ui, uj, uk) = (−1, 0, 0) as (ui, uj, uk) = (vivk − 1, vjvk, vk), then the exceptional curve ei is
given by vk = vj − (vi + 1)2 = 0. Parametrize ei as (vi, vj , vk) = (−(t + 1)/t, t−2, 0), where
qi corresponds to t = ∞. Then g̃2j acts on ei by the shift t 7→ t + 1. Similarly g̃2j acts on ek
as a parabolic transformation fixing qk only. By symmetry, g̃
i and g̃
k act on ej as parabolic
transformations fixing qj only. Notice that the exceptional curve e0 carries rational Riccati
solutions, while ej − {qj} carries Riccati solutions of infinte period. Thus we have
F̃ixj(θ) = ℓ̃
e0, P̃er
j(θ;n) = ∅ (n > 1). (58)
Example 7.8 (A
) Consider the W̃ (D
4 )-stratum of type A
1 , where θ = (0, 0, 0,−4) and
Fixj(θ) = ℓ
j ∐ℓ−j . In this case the surface S(θ) has four singularities of type A1 at (xi, xj, xk) =
(2εi, 2εj, 2εk) ∈ {±2}3 with εiεjεk = −1. Blow up at these points to obtain a minimal resolution
as in (15). Let eεiεjεk be the exceptional line over (xi, xj, xk) = (2εi, 2εj, 2εk) and ℓ̃
j be the
strict transform of ℓ
j . Moreover let p
εiεjεk denote the intersection point of the lines eεiεjεk
and ℓ̃
j (see Figure 17). Then the lifted transformation g̃
j : S̃(θ) 	 acts on the exceptional
line eεiεjεk ∼= P1 as a Möbius transformation. It is a parabolic transformation with the only
fixed point pεiεjεk . Let us check this for (εi, εj, εk) = (−1,−1,−1). The blowing-up of C3 at
(xi, xj, xk) = (−2,−2,−2) is described by xi = uiuj − 2, xj = uj − 2, xk = ujuk − 2, in terms
of coordinates (ui, uj, uk) around (0, 0, 0). Then the exceptional line e
−−− is represented by the
equations uj = 0 and (ui − uk)2 − 2(ui + uk) + 1 = 0 and hence it is parametrized as
, uj = 0, uk =
where the fixed point p−−− corresponds to t = ∞. Then we can check that g̃2j acts on the line
e−−− as the translation t 7→ t+4, as desired. Thus the only fixed points of g̃2j on the exceptional
set E(θ) are the four points pεiεjεk with εiεjεk = −1 and there are no periodic points, so that
F̃ixj(θ) = ℓ̃
j ∐ ℓ̃−j , P̃er
j(θ;n) = ∅ (n > 1). (59)
8 Power Geometry
We apply the method of power geometry [5, 6, 7] to construct as many algebraic branch solutions
to PVI(κ) as possible around each fixed singular point. Basically we can follow the arguments
of [7]. However, while the attention of [7] is restricted to generic parameters, we require a
thorough treatment of all parameters, where much ampler varieties of patterns are present.
Moreover, the way in [7] of representing the parameters of Painlevé VI is not convenient for our
purpose. So we have to redevelop the necessary arguments on power geometry from scratch.
In view of Remark 5.1, it is sufficient to work around the origin z = 0. In order to apply
the method in [5, 6, 7], we reduce the system (1) into a single second-order equation. If
(q, p) = (q(z), p(z)) is a solution to system (1) such that q 6≡ 0, 1, z, ∞, then we solve the first
equation of system (1) with respect to p = p(z) to obtain
z(z − 1)q′ + κ1q1qz + (κ2 − 1)q0q1 + κ3q0qz
2q0q1qz
. (60)
Substituting this into the second equation yields the single second-order equation
z − 1 +
q0q1qz
2z2(z − 1)2
κ24 − κ21
+ κ23
z − 1
+ (1− κ22)
z(z − 1)
Multiply equation (61) by 2z2(z − 1)2q0q1qz and move its right-hand side to the left to obtain
P (z, q) = 0, (62)
where P (z, q) is a polynomial of (z, q, q′, q′′), that is, a differential sum of (z, q), whose explicit
formula is omitted here but can be found in [7]. Therefore system (1) is equivalent to equation
(62) together with (60) except for the possible solutions such that q ≡ 0, 1, z, ∞. A simple
check shows that the Newton polygon of equation (62) is given as in Figure 18, where there are
four patterns according as the parameters κ1 and κ4 are zero or not.
First we search for a holomorphic solution germ q = q(z) to equation (62) around z = 0.
We have only to construct formal power series solutions of the form
q = czr + (higher order terms), (r, c) ∈ Z× C×, (63)
0 (3, 0)
(0, 3)
(0, 6)
(3, 3)
Case κ1 6= 0, κ4 6= 0.
(0, 3)
(0, 6)
(3, 3)
(3, 2)(1, 2)
Case κ1 = 0, κ4 6= 0.
0 (3, 0)
(0, 3)
(3, 3)Γ0
(0, 4) (2, 4)
Case κ1 6= 0, κ4 = 0.
(0, 3)
(3, 3)Γ0
(0, 4) (2, 4)
(3, 2)(1, 2)
Case κ1 = 0, κ4 = 0.
Figure 18: Newton polygon for Painlevé VI
0 (4, 0)
(0, 3)
(0, 6)
(2, 1)
(6, 0) 0 (3, 0)
(0, 1)
(0, 6)
(3, 3)
(1, 0)
Figure 19: Newton polygons for Lemma 8.1 (left) and Lemma 8.2 (right)
since any formal power series solution to equation (62) is convergent [10, 11]. Then it follows
from (60) that the associated formal Laurent series for p = p(z) is also convergent.
In order to construct formal solutions (63), we consider the truncations along the edges
Γ1 and Γ0 of the Newton polygons in Figure 18. We see that Γ1 and Γ0 have outer normal
vectors (p1, p2) = (−1,−1) and (p1, p2) = (−1, 0), whose slopes are p2/p1 = 1 and p2/p1 = 0
respectively. Thus the edges Γ1 and Γ0 correspond to the exponents r = 1 and r = 0 respectively.
The truncation of P = P (z, q) along the edge Γ1 is given by
P1 = −2zq2q′ + 2z2q(q′)2 − 2z2q2q′′ + (κ21 − κ22 + 1)zq2 − z3(q′)2 + 2z3qq′′
−2κ21z2q + κ21z3.
Substituting q = cz into equation P1 = 0 yields c = κ1/(κ1 + εκ2) with any sign ε ∈ {±1}.
Similarly, the truncation of P = P (z, q) along the edge Γ0 is given by
P0 = −2zq2q′ + 2z2q(q′)2 − 2z2q2q′′ + (κ23 − κ24)q4 + 2zq3q′ − 3z2q2(q′)2
+2z2q3q′′ + 2κ24q
5 − κ24q6.
Substituting q = c into equation P0 = 0 yields c = (κ4 + εκ3)/κ4 with any sign ε ∈ {±1}.
Lemma 8.1 If κ1 + κ2 6∈ Z, then there exists a holomorphic solution around the origin z = 0,
κ1 + κ2
+ κ1κ2
ak,+(κ) z
k, p = κ0(κ0 + κ4)
bk,+(κ) z
k, (64)
depending holomorphically on κ ∈ K with κ1 + κ2 6∈ Z. Similarly, if κ1 − κ2 6∈ Z, then there
exists a meromorphic solution around the origin z = 0,
κ1 − κ2
+ κ1κ2
ak,−(κ) z
k, p =
κ1 − κ2
bk,−(κ) z
k, (65)
depending holomorphically on κ ∈ K with κ1 − κ2 6∈ Z.
Proof. Substituting q = κ1z(κ1 + εκ2)
−1 + κ1κ2Q with ε ∈ {±1} into equation (62) yields
(κ1 + εκ2)
z, κ1z(κ1 + εκ2)
−1 + κ1κ2Q
= κ21κ
2 p(z, Q),
where p(z;Q) is a differential sum of (z, Q) with coefficients in C[κ] whose Newton polygon is
given as in Figure 19 (left). The vertex (2, 1) carries the linear differential expression
LεQ = 2ε(κ1 + εκ2)4x2{x2Q′′ − xQ′ − (κ1 + εκ2 + 1)(κ1 + εκ2 − 1)Q},
while the vertex (4, 0) carries the monomial (κ1 + εκ2)
2{(κ1 + εκ2)2 + κ23 − κ24 − 1}x4. The
corresponding characteristic polynomial is given by
vε(k) = 2ε(κ1 + εκ2)
4(k − 1− κ1 − εκ2)(k − 1 + κ1 + εκ2).
Hence 1 + |κ1 + εκ2| is the unique critical value of the problem. If it is not an integer, then
the coefficients ak,ε(κ) of the expansions (64) and (65) are determined uniquely and recursively.
By substituting the resulting power series q = q(z) into equation (60), the Laurent series for
p = p(z) is uniquely determined as in (64) and (65). As is mentioned earlier, the formal
solutions (64) and (65) so obtained are convergent. ✷
Lemma 8.2 Assume that κ4 is nonzero. If κ4 + κ3 6∈ Z, then there exists a holomorphic
solution germ around the origin z = 0,
κ4 + κ3
ak,+(κ) z
k, p = −κ4κ0
bk,+(κ) z
k, (66)
depending holomorphically on κ ∈ K with κ4 6= 0 and κ4 + κ3 6∈ Z. Similarly, if κ4 − κ3 6∈ Z
then there exists a holomorphic solution germ around the origin z = 0,
κ4 − κ3
ak,−(κ) z
k, p = −κ4(κ0 + κ4)
bk,−(κ) z
k, (67)
depending holomorphically on κ ∈ K with κ4 6= 0 and κ4 − κ3 6∈ Z.
Proof. Substituting q = κ−14 (κ4 + εκ3) + κ
4 κ3Q into equation (62) yields
z, κ−14 (κ4 + εκ3) + κ
4 κ3Q
= κ23 p(z;Q),
where p(z;Q) is a differential sum of (z, Q) with coefficients in C[κ]. The vertex (0, 1) carries
the linear differential expression LεQ = 2(κ4 + εκ3)2{x2Q′′ + xQ′ − (κ4 + εκ3)2Q}, while the
vertex (1, 0) carries the monomial (κ4 + εκ3)
2{1 + κ21 − κ22 − (κ4 + εκ3)2}x. The corresponding
characteristic polynomial is given by vε(k) = 2(κ4 + εκ3)
2{k − (κ4 + εκ3)}{k + (κ4 + εκ3)}.
Hence |κ4 + εκ3| is the unique critical value of the problem. If it is not an integer, then the
coefficients ak,ε(κ) of expansions (66) and (67) are determined uniquely and recursively. Then
substituting the resulting series for q = q(z) into equation (60) yields the Laurent series for
p = p(z) as in (66) and (67). The formal solutions so obtained are convergent. ✷
The solutions in Lemmas 8.1 and 8.2 are essentially constructed in [7, 23]. We construct
more particular solutions for the parameters on various strata of higher codimensions.
0 (4, 0)
(0, 3)
(0, 6)
(2, 1)
Figure 20: Newton polygon for Lemma 8.3
Lemma 8.3 (A
and A
) If κ1 = κ2 = 0, then there exists a 1-parameter family of holo-
morphic solution around the origin z = 0,
t+ (1− t)(1− z)κ4 + t(1− t)κ0(κ0 + κ3)
ak(t; κ) z
p = κ0(κ0 + κ4)
bk(t; κ) z
depending on t ∈ C, where a2(t; κ) = 2, b0(t; κ) = 1 and the remaining coefficients ak(t; κ),
k ≥ 3, and bk(t; κ), k ≥ 1, are polynomials of (t, κ3, κ4) determined uniquely and recursively.
Proof. We put R(z; t) = tz{t+(1−t)(1−z)κ4}−1. Substituting q = R(z; t)+t(1−t)κ0(κ0+κ3)Q
into equation (62) and multiplying the result by {t+ (1− t)(1− z)κ4}4 yield
p(z, Q; t) := {t+ (1− t)(1− z)κ4}4P (z, R(z; t) + t(1− t)κ0(κ0 + κ3)Q)
= 2t2(1− t)2κ0(κ0 + κ3){LQ+ g(z, Q; t) + h(z)} = 0,
where LQ = z2{z2Q′′ − zQ′ + Q} and h(z) = −2z4(1 − z)2κ4+1. The Newton polygon of (69)
is given as in Figure 20, where the terms LQ and h(z) correspond to the vertex (2, 1) and the
horizontal infinite edge emanating from the vertex (4, 0) respectively, and the remaining term
g(z, Q; t) corresponds to the remaining part of the polygon. Since the characteristic equation
of LQ is (k − 1)2 = 0 having the unique root k = 1, the coefficients ak(t; κ), k ≥ 2, in (68)
are determined uniquely and recursively. Here the leading coefficient a2(t; κ) is found to be
a2(t; κ) = 2 by substituting Q = a2(t; κ)z
2 into the truncation LQ − 2z4 = 0 of equation
(69) along the edge connecting the vertices (2, 1) and (4, 0). Substituting the resulting series
q = q(z) into (60) we have p = p(z) as in (68). ✷
(0, 6)
(0, 1)
(2, 0)
Figure 21: Newton polygon for Lemma 8.4
Lemma 8.4 (A3) If κ0 = κ1 = κ2 = 0, κ3 + κ4 = 1, then there exists a 1-parameter family of
holomorphic solutions around the origin z = 0,
1− (1− z)κ4 + tκ3z + tκ3κ4
ak(t; κ) z
k, p = −tκ34
bk(t; κ) z
k, (70)
depending on t ∈ C, where a2(t; κ) = tκ3, b1(t; κ) = 1 and the coeffcients ak(t; κ), k ≥ 2, and
bk(t; κ), k ≥ 1, are polynomials of (t, κ4) determined uniquely and recursively.
Proof. Put R(z) = z{1− (1− z)κ4}−1. Substituting q = R(z) + tκ3z + tκ3κ4Q into (62) yields
P (z, R(z) + tκ3z + tκ3κ4Q) = tκ
4R(x)
4 p(z, Q; t),
where p(z, Q; t) is a differential sum of (z, Q) with coefficients in C[t, κ4] whose Newton polygon
is given as in Figure 21. Especially the vertex (0, 1) carries the linear differential expression
LQ = 2(z2Q′′ + zQ′ −Q), whose characteristic polynomial is 2(k − 1)(k + 1), while the vertex
(2, 0) carries the monomial −6tκ3z2. Since the critical values k = ±1 are smaller than 2, the
coefficients ak(t; κ), k ≥ 2, in (70) are determined uniquely and recursively, where the leading
coefficient a2(t; κ) is found to be tκ3. The rest of the proof is similar to that in Lemma 8.3. ✷
Lemma 8.5 (D4) If κ0 = κ1 = κ2 = κ3 = 0 and κ4 = 1, then there exists a 1-parameter
family of holomorphic solution germs around the origin z = 0,
q = 1 + t
ak(t) z
k, p =
(1− z) log(1− z) + t
bk(t) z
k, (71)
depending on a parameter t ∈ C, where a1(t) = b1(t) = 1 and the remaining coeffieicents ak(t)
and bk(t), k ≥ 2, are polynomials of t determined uniquely and recursively.
0 (3, 0)
(0, 2)
(0, 6)
(1, 1)
(6, 0) 0
(2, 1)(0, 1)
(0, 4)
(2, 3)
(1, 0)
Figure 22: Newton polygons for Lemma 8.5 (left) and Lemma 8.6 (right)
Proof. Substituting q = 1+ tz + tQ into equation (62) yields P (z, 1+ tQ) = t2p(z, Q; t), where
p(z, Q; t) is a differential sum of (z, Q) with coefficients in C[t] whose Newton polygon is given
as in Figure 22 (left). Consider the edge connecting (1, 1) and (3, 0). The vertex (1, 1) carries
the differential monomial LQ = 2z3Q′′, whose characteristic polynomial is k(k − 1), while the
vertex (3, 0) carries the monomial −4tz3. Put a1(t) = 1. Since the critical values k = 0, 1 are
smaller than 2, the coefficients ak(t), k ≥ 2, in (71) are determined uniquely and recursively.
Substituting the resulting series q = q(z) into (60) we have p = p(z) as in (71). Here the term
z/{(1 − z) log(1 − z)} is singled out from p = p(z), because putting t = 0 yields the special
solution q ≡ 1 and p = z/{(1− z) log(1− z)} (see also Lemma 9.4). ✷
Lemma 8.6 (A
) Let κ0 = 1/2 and κ1 = κ2 = κ3 = κ4 = 0. Then there exists a 1-parameter
family of holomorphic solutions around the origin z = 0 depending on a parameter t ∈ C,
q = tz + t(1− t)
ak(t) z
k, p =
bk(t) z
k, (72)
where the coefficients ak(t) and bk(t) are polynomials of t beginning with a2(t) = 1/2 and
b0(t) = 1/4. Moreover there is another 1-parameter family of solutions around z = 0,
ck(t) z
k, p = t
dk(t) z
k, (73)
depending on t ∈ C, where ck(t) and dk(t) are polynomials of t beginning with c1(t) = 1/2 and
d0(t) = −1/2. For t = 0, formula (73) represents the solution such that q ≡ ∞ and p ≡ 0.
Proof. We only derive (73), as (72) is derived in a similar manner. Substituting q = t−1 +
t−1(t−1)Q into (62) yields P (z, t−1+t−1(t−1)Q) = t−4(t−1)2z−1p(z, Q; t), where p(z, Q; t) is a
differential sum of (z, Q) with coefficients in C[t] whose Newton polygon is given as in Figure 22
(right). Consider the edge connecting (0, 1) and (1, 0). The vertex (0, 1) carries the differential
sum LQ = −2(zQ′ + z2Q′′), whose characteristic polynomial is −2k2, while the vertex (1, 0)
carries the monomial z. Since the critical value k = 0 is smaller than 1, the coefficients ck(t),
k ≥ 1, in (73) are determined uniquely and recursively, where the leading term is c1(t) = 1/2.
Substituting the resulting series q = q(z) into (60) we have p = p(z) as in (73). ✷
So far we have considered the edges Γ1 and Γ0 of the Newton polygon in Figure 18 and
constructed meromorphic solutions around the origin z = 0. Now let us consider the vertex (0, 3)
of the polygon, which gives rise to algebraic branch solutions around z = 0. The truncation of
the differential sum P = P (z, q) at the vertex (0, 3) is given by
P3 = −2zq2q′ + 2z2q(q′)2 − 2z2q2q′′.
Its charactersistic polynomial χ(r) is defined by substituting q = zr into P3(z, q) and dividing
the result by q3 = z3r. In the present situation we see that χ(r) is identically zero; χ(r) ≡ 0.
The normal cone of the vertex (0, 3) is U3 = { (p1, p2) ∈ R2 : p1 < 0, 0 < r = p2/p1 < 1 }.
Thus the truncated solutions at the vertex (0, 3) are q = tzr for an arbitrary 0 < r < 1 and
t ∈ C×. The Fréchet derivative with respect to q at the truncated solution q = tzr is given by
L3Q = −2t2z2r{z2Q′′ + (1− 2r)zQ′ + r2Q}.
The corresponding characteristic equation is given by v3(k) := −2t2(k − r)2 = 0, which has
the only root k = r. Let n be any integer greater than 1. In order to search for an algebraic
n-branch solution around z = 0, we take r = m/n for any integer 0 < m < n coprime to n,
and consider a formal Puiseux series solution of the form
q = tzm/n +
ν=m+1
aν(t) z
Since the characteristic equation v3(k) = 0 has no roots such that k > m/n, the coefficients
aν = aν(t) can be determined uniquely and recursively for any given initial coefficient t ∈ C×.
The convergence of the formal solution and its holomorphic dependence on parameters follow
easily if we rewrite the equation (62) in terms of the new independent variable ζ = z1/n and
apply the convergence arguments in [10, 11]. Thus we have established the following lemma.
Lemma 8.7 For any integer n > 1, there exist ϕ(n) mutually disjoint 1-parameter families of
n-branch solution germs to PVI(κ) around the origin z = 0,
q(z) = tzm/n +
ν=m+1
aν(n,m, t; κ) z
p(z) =
m+ n(κ1 + κ2 − 1)
z−m/n +
ν=−m+1
bν(n,m, t; κ) z
where the discrete parameter m ranges over all integers 0 < m < n coprime to n and the
continuous parameter t takes any value of the punctured complex line C×.
Remark 8.8 The family (74) contains no Riccati solutions, even if κ ∈ Wall. This will be
shown in the proof of Lemma 9.14.
9 Injection Implies Surjection
We establish Theorem 1.3 based on the main idea described in §1. Due to the S3-symmetry
permuting the three fixed singular points, it suffices to work around z = 0 (see Remark 5.1).
As a preliminary we begin by constructing some Riccati solutions to equation (1). Assume
that κ0 = 0 so that κ1 + κ2 + κ3 + κ4 = 1. Then the second equation of system (1) has the null
solution p(z) ≡ 0. Substituting this into the first equation yields the Riccati equation
z(z − 1)q′ + κ1q1qz + (κ2 − 1)q0q1 + κ3q0qz = 0.
If κ4 is nonzero, then the change of dependent variable
z(1 − z)
log{(1− z)−κ4f}
transfers the Riccati equation to the Gauss hypergeometric equation
z(1− z)f ′′ + {(1− κ3 − κ4)− (κ2 − κ4 + 1)z}f ′ + κ2κ4f = 0. (75)
Next assume that κ0 = κ1 = 0 so that κ2 + κ3 + κ4 = 1. In this case there is another type
of Riccati solution to the system (1). The first equation of the system (1) has the null solution
q(z) ≡ 0. Substituting this into the second equation yields the Riccati equation
z(z − 1)p′ + zp2 + (κ2 − 1 + κ3z)p = 0.
Then change of independent variable p = (z − 1) d
log g takes it to the linear equation
z(1 − z)g′′ + {(1− κ2)− (κ3 + 1)z}g′ = 0. (76)
Lemma 9.1 (A1) Assume that κ0 = 0, κ1 + κ2 + κ3 + κ4 = 1, κ1 + κ2 6∈ Z, κ3 + κ4 6∈ Z and
κ1κ4 6= 0. Then system (1) has two single-valued Riccati solutions around the origin z = 0,
(a) q =
κ1 + κ2
+O(κ1z
2), p ≡ 0,
(b) q =
κ3 + κ4
+O(κ3z), p ≡ 0.
Proof. The solutions (a) and (b) are obtained from two linearly independent solutions
2F1(κ2,−κ4, 1− κ3 − κ4; z), zκ3+κ42F1(κ2 + κ3 + κ4, κ3, κ3 + κ4 + 1; z),
of equation (75) repsectively. They also come from solutions (64) and (66) respectively. ✷
Lemma 9.2 (A2) Assume that κ0 = κ1 = 0, κ2 + κ3 + κ4 = 1, κ2 6∈ Z, κ3 6= 1 and κ4 6= 0.
Then system (1) has three single-valued Riccati solutions around the origin z = 0,
(a) q ≡ 0, p ≡ 0,
(b) q =
κ3 + κ4
+O(z), p ≡ 0,
(c) q ≡ 0, p = −κ2(κ3 − 1)
κ2 + 1
+O(z).
Proof. The solutions (a) and (b) just come from (a) and (b) of Lemma 9.1 respectively, while
solution (c) is obtained from the solution zκ22F1(κ2, κ2 + κ3, κ2 + 1; z) of equation (76). ✷
Lemma 9.3 (A3) Assume that κ0 = κ1 = κ2 = 0, κ3 + κ4 = 1 and κ4 6= 0. Then system (1)
admits a 1-parameter family of single-valued Riccati solutions around the origin z = 0,
q(z; s) =
s0 + s1(1− z)κ4
, p(z; s) ≡ 0, (s = [s0 : s1] ∈ P1). (77)
Proof. The hypergeometric equation (75) becomes (1 − z)f ′′ − κ3f ′ = 0, whose nontrivial
solutions are given by f(z) = s0 + s1(1− z)κ4 with (s0, s1) ∈ C2 − {(0, 0)}. The corresponding
Riccati solutions are the 1-parameter family of single-valued solutions q = q(z; s) as in (77). ✷
Lemma 9.4 (D4) Assume that κ0 = κ1 = κ2 = κ3 = 0 and κ4 = 1.
(1) System (1) admits a 1-parameter family of rational Riccati solutions
q(z; s) =
s0 + s1(1− z)
, p(z; s) ≡ 0, (s = [s0 : s1] ∈ P1). (78)
(2) System (1) admits a 1-parameter family of single-valued Riccati solutions around z = 0,
q(z; t) ≡ 1, p(z; t) = t0z
(1− z){t0 log(1− z) + t1}
, (t = [t0 : t1] ∈ P1). (79)
Proof. In this case (77) gives the 1-parameter family of rational solutions (78). Moreover the
first equation of system (1) admits a constant solution q(z) ≡ 1. Substituting this into the
second equation yelds the Riccati equation z(z−1)p′+(1− z)p2+ p = 0. Change of dependent
variable p = −zf ′/f takes this into the linear equation (1 − z)f ′′ − f ′ = 0, whose nontrivial
solutions are given by f = t0 log(1 − z) + t1 with (t0, t1) ∈ C2 − {(0, 0)}. Thus the Riccati
equation has the 1-parameter family of single-valued solutions p = p(z; t) as in (79). ✷
Now we proceed to the proof of Theorem 1.3. From now on we fix the indices as (i, j, k) =
(3, 1, 2) in accordance with the choice of indices in §8. First we treat the fixed point case.
Lemma 9.5 The set F̃ixj(θ) is exhausted by meromorphic solutions around z = 0.
Proof. Case-by-case check based on the “injection-implies-surjection” principle described in §1.
Example 9.6 (∅) We combine the results of Example 7.1, Lemmas 8.1 and 8.2. A key ob-
servation is that Lemmas 8.1 and 8.2 give us as many meromorphic solutions around z = 0
as the cardinality of the set F̃ixj(θ) in (49). For example, if P̃ (bi, b4; bj , bk) ∈ F̃ixj(θ), then
the existing and smoothness conditions for it (see Table 3) makes it possible to apply Lemma
8.2 to conclude that the meromorphic solution (66) exists corresponding to the fixed point
P̃ (bi, b4; bj, bk). Thus the set F̃ixj(θ) is exhausted by meromorphic solutions around z = 0.
Example 9.7 (A1) We combine the results of Example 7.2, Lemmas 8.1, 8.2 and 9.1. First we
notice that the two single-valued Riccati solutions in Lemma 9.1 correspond to the two Riccati
fixed points F̃ix
j(θ) = {p, q} in (50). On the other hand, for the same reason as in Example 9.6,
formulas (65) and (67) in Lemmas 8.1 and 8.2 give us as many meromorphic solutions around
z = 0 as the cardinality of smooth fixed points F̃ix
j(θ) = {{P̃ (bi, b−14 ; bj , bk), P̃ (bj , b−1k ; bi, b4)}}
in (50). Thus the set F̃ixj(θ) is exhausted by meromorphic solutions around z = 0.
Example 9.8 (A2) We combine the results of Example 7.3, Lemmas 8.2 and 9.2. As (51)
shows, the set F̃ixj(θ) consists of the four points p0, p+, p− and P̃ (bj , b
k ; bi, b4). On the other
hand, we have the three single-valued Riccati solutions of Lemma 9.2 and one non-Riccati
holomorphic solution (67). Clearly, the three Riccati solutions correspond to the points p0,
p+ and p−, while the non-Riccati solution corresponds to the remaining point P̃ (bj , b
k ; bi, b4).
Thus any single-valued solution around z = 0 is a meromorphic solution.
Example 9.9 (A
) We combine the results of Example 7.4, Lemmas 8.2 and 8.3. First we
consider the W̃ (D
4 )-stratum of type (A
1 )i. The C-parameter family (68) of holomorphic
solutions injects into the line ℓ̃+j ≃ C in (53), so that we have an injection C →֒ C. Since this
injection is holomorphic, it must be a surjection. Thus the line ℓ̃+j is exhausted by the family
(68). Moreover we have the two holomorphic solutions (66) and (67), which do not lie in the
family (68). Thus they must correspond to the points P̃ (bi, b4; bj, bk) and P̃ (bi, b
4 ; bj , bk) in
(53). So on the stratum of type (A⊕21 )i the set F̃ixj(θ) is exhausted by meromorphic solutions
around z = 0. Next we consider the W̃ (D
4 )-strata of types (A
1 )j and (A
1 )k. On these
strata the equality F̃ixj(θ) = F̃ix
j(θ) in (54) implies that any single-valued solution around
z = 0 is a Riccati and hence meromorphic solution.
Example 9.10 (A3) We combine the results of Example 7.5, Lemmas 8.2, 8.4 and 9.3. First
we consider the W̃ (D
4 )-stratum of type (A
1 )i. The P
1-family (77) of single-valued Riccati
solutions exactly corresponds to the exceptional curve e0 ≃ P1 in (55). So the C-family (70) of
holomorphic solutions must inject into the line ℓ̃+j ≃ C in (55). Since this injection C →֒ C is
holomorphic, it must be a surjection. Hence ℓ̃+j is exhausted by the family (70). Moreover there
is the holomorphic solution (67), which must correspond to the point P̃ (bi, b
4 ; bj, bk) in (55).
Thus on the stratum of type (A⊕21 )i the set F̃ixj(θ) is exhausted by meromorphic solutions
around z = 0. Next we consider the W̃ (D
4 )-strata of types (A
1 )j and (A
1 )k. On these
strata the equality F̃ixj(θ) = F̃ix
j(θ) in (56) implies that any single-valued solution around
z = 0 is a Riccati and hence meromorphic solution.
Example 9.11 (A
) We combine the results of Example 7.6 and Lemma 8.3. As (57) shows,
F̃ixj(θ) has only one line component ℓ̃
j ≃ C. Hence the C-family (68) of holomorphic solu-
tions must inject into this line, so that we have an inclusion C →֒ C. Since this injection is
holomorphic, it must be a surjection. Thus ℓ̃+j is exhausted by the family (68).
Example 9.12 (D4) We combine the results of Example 7.7, Lemmas 8.5 and 9.4. The P
family (78) of rational Riccati solutions corresponds to the exceptional curve e0 in (58), while
the P1-family (79) of single-valued Riccati solutions corresponds to the exceptional curve ej
there. Hence C-family (71) of holomorphic solutions, which is different from (78) and (79),
must inject into the line ℓ̃+j ≃ C in (58). Since this injection C →֒ C is holomorphic, it must
be a surjection. Thus ℓ̃+j is exhausted by the family (71).
Example 9.13 (A
) We combine the results of Example 7.8 and Lemma 8.6. In view of
F̃ixj(θ) = ℓ̃
j ∐ ℓ̃−j , the C-family (72) of holomorphic solutions injects into the line ℓ̃εj ≃ C for
some sign ε ∈ {±1}. So we have an injection C →֒ C. Since this injection is holomorphic, it
must be a surjection, so that ℓ̃εj is exhausted by the family (72). Then the other C-family (73)
of holomorphic solutions injects into the remaining line ℓ̃−εj ≃ C. So we have another injection
C →֒ C. Since this injection is holomorphic, it must be a surjection. Hence F̃ixj(θ) = ℓ̃+j ∐ ℓ̃−j
is exhausted by the families (72) and (73). The proof of Lemma 9.5 is now complete. ✷
Finally we argue the periodic point case using the “injection-implies-surjection” principle.
Lemma 9.14 For any n > 1 the set P̃erj(θ;n) is exhausted by algebraic n-branch solutions
around z = 0.
Proof. We combine Lemmas 6.6 and 8.7. First we consider the generic case where κ ∈ K−Wall,
namely, where θ = rh(κ) is such that ∆(θ) 6= 0. In this case there is no Riccati locus and hence
P̃erj(θ;n) = P̃er
j (θ;n), which is biholomorphic to the disjoint union of ϕ(n) copies of C
Lemma 6.6. On the other hand, by Lemma 8.7, there are ϕ(n) mutually disjoint C×-parameter
families of algebraic n-branch solutions around z = 0 as in (74). Number these families from
1 to ϕ(n). The first family injects into a (unique) connected component (≃ C×) of P̃erj(θ;n),
which we call the first component, and we have an injection C× →֒ C×. Since this injection is
holomorphic, it must be a surjection and hence the first component is exhausted by the first
family. Consider the second family of solutions and the corresponding second component of
P̃erj(θ;n). Notice that the second component is different from the first one, because the first
component is already occupied by the first family and so it cannot contain the second family.
For the same reason as above, the second component is exhausted by the second family. Since
the families and the components have the same cardinality ϕ(n), we can repeat this argument
to conclude that P̃erj(θ;n) is exhausted by the ϕ(n) families of algebraic n-branch solutions.
Next we consider the case where κ ∈ Wall, namely, where the Riccati part P̃er
j(θ;n) may
appear. Since the lemma is trivial for the Riccati part, we have only to consider the non-Riccati
part P̃er
j(θ;n). The argument proceeds just in the same manner as in the last paragraph, once
we show that the family of solutions in (74) contains no Riccati solutions (see Remark 8.8). To
see this, we consider the family S̃ → Θ of surfaces S̃(θ) parametrized by θ ∈ Θ and put
j (n) =
j (θ;n), E =
E(θ),
where E(θ) is the exceptional set in S̃(θ). (Precisely speaking, the parameter space Θ should
be replaced by a finite covering of it to get a simultaneous minimal resolution.) Then P̃er
j (n)
and E are closed subsets of S̃ which are disjoint by Lemma 4.5. Now we look at the family
of solutions in (74). It depends continuously on κ ∈ K. Take any point κ∗ ∈ Wall and let
K − Wall ∋ κ → κ∗. For any κ ∈ K −Wall, the family at κ is contained in P̃er
j(θ;n) with
θ = rh(κ) and hence in P̃er
j(n). Taking the limit κ → κ∗, we see that the family at κ∗ is
contained in P̃er
j (n), hence in P̃er
∗;n) with θ∗ = rh(κ∗). Since P̃er
∗;n) is disjoint from
E(θ∗), the family at κ∗ contains no Riccati solutions. Therefore the proof is complete. ✷
Now the local statement of Theorem 1.3 around a fixed singular point, say z = 0, is an
immediate consequence of Lemmas 9.5 and 9.14. At the same time all the finite branch solutions
around z = 0 have been classified up to Bäcklund transformations. The global statement about
algebraic solutions follows readily from the local statements around z = 0, 1, ∞, together with
the analytic Painlevé property on Z = P1 − {0, 1,∞}. The proof of Theorem 1.3 is complete.
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	Introduction
	Phase Space
	Riemann-Hilbert Correspondence
	Dynamics on Cubic Surface
	Bäcklund Transformations
	Fixed Points and Periodic Points
	Case-by-Case Study
	Power Geometry
	Injection Implies Surjection
