Arata Komyo

AND Frank Loray
email: frank.loray@univ-rennes1.fr

Masa-Hiko Saito
email: mhsaito@ba.kobegakuin.ac.jp

MODULI SPACE OF IRREGULAR RANK TWO PARABOLIC BUNDLES OVER THE RIEMANN SPHERE AND ITS COMPACTIFICATION

Keywords: . 2010 Mathematics Subject Classification. Primary 14H60, Secondary 14D20 14Q10 Parabolic bundle, refined parabolic bundle, projective line, elementary transformation, moduli space, weak del Pezzo surface

In this paper, we study rank 2 (quasi) parabolic bundles over the Riemann sphere with an effective divisor and these moduli spaces. First we consider a criterium when a parabolic bundle admits a unramified irregular singular parabolic connection. Second, to give a good compactification of the moduli space of semistable parabolic bundles, we introduce a generalization of parabolic bundles, which is called refined parabolic bundles. Third, we discuss a stability condition of refined parabolic bundles and define elementary transformations of the refined parabolic bundles. Finally, we describe the moduli spaces of refined parabolic bundles when the dimensions of the moduli spaces are two. These are related to geometry of some weak del Pezzo surfaces.

Introduction

In this paper, we investigate the moduli space of rank 2 (quasi) parabolic bundles over the Riemann sphere with an effective divisor D and extend some results obtained in [START_REF] Loray | Lagrangian fibrations in duality on moduli spaces of rank 2 logarithmic connections over the projective line[END_REF] when the effective divisor D is reduced. Let r and ν be positive integers and C be a smooth projective curve over C. Fix points t 1 , t 2 , . . . , t ν on C, and set

D = n 1 [t 1 ] + • • • + n ν [t ν ].
In general, a parabolic bundle (E, l) over (C, D) is a pair where E is a rank r vector bundle over C, l = {l j i } 0≤j≤r-1 1≤i≤ν , which are filtrations

E| ni[ti] = l 0 i ⊃ l 1 i ⊃ • • • ⊃ l r-1 i ⊃ l r i = 0 by free O ni[ti] -modules such that l j i /l j+1 i ∼ = O ni[ti]
for any i, j. Parabolic (vector) bundles over an irreducible smooth complex projective curve with a reduced effective divisor were introduced in Mehta-Seshadri [START_REF] Mehta | Moduli of vector bundles on curves with parabolic structures[END_REF]. To construct a good moduli space, a stability condition for the parabolic weights α was introduced. The moduli spaces of α-semistable parabolic bundles were constructed in [START_REF] Mehta | Moduli of vector bundles on curves with parabolic structures[END_REF], [START_REF] Bhosle | Parabolic vector bundles on curves[END_REF], and [START_REF] Bhosle | Moduli space of parabolic vector bundles on a curve[END_REF]. The effect on the moduli space of varying the parabolic weights was studied in [START_REF] Boden | Variations of moduli of parabolic bundles[END_REF], [START_REF] Bauer | Parabolic bundles, elliptic surfaces and SU(2)-representation spaces of genus zero Fuchsian groups[END_REF], [START_REF] Mukai | Finite generation of the nagata invariant rings in A-D-E cases[END_REF], and [START_REF] Moon | Birational geometry of the moduli space of rank 2 parabolic vector bundles on a rational curve[END_REF]. This effect is interesting from the point of view of the birational geometry (in the sense of Mori's program). A generalization of parabolic bundles was studied in [START_REF] Bhosle | Generalised parabolic bundles and applications to torsionfree sheaves on nodal curves[END_REF] and [START_REF] Bhosle | Generalized parabolic bundles and applications[END_REF]. Here the (generalized) parabolic bundles are defined over an irreducible smooth complex projective curve with an effective divisor, which is not necessary reduced. A stability condition of such parabolic bundles was introduced and the moduli spaces of α-semistable parabolic bundles were constructed in [START_REF] Bhosle | Generalized parabolic bundles and applications[END_REF]. Finally a more general notion of parabolic bundles was studied by Yokogawa ([27] and [START_REF] Yokogawa | Infinitesimal deformation of parabolic Higgs sheaves[END_REF]).

Usually, a parabolic bundle consists of a vector bundle E, filtrations l = {l j i } 0≤j≤r-1 1≤i≤ν

, and parabolic weights α. But in this paper, we call a pair (E, l) parabolic bundle and call the filtration l a parabolic structure. Parabolic structures are important for study of geometry of isomonodromic deformations (or Painlevé type equations). For example, parabolic connections were studied in [START_REF] Inaba | Moduli of stable parabolic connections, Riemann-Hilbert correspondence and geometry of Painlevé equation of type VI[END_REF], [START_REF] Inaba | Moduli of stable parabolic connections, Riemann-Hilbert correspondence and geometry of Painlevé equation of type VI. II. Moduli spaces and arithmetic geometry[END_REF], and [START_REF] Inaba | Moduli of parabolic connections on curves and the Riemann-Hilbert correspondence[END_REF]. A parabolic connection is a pair of a connection and a parabolic structure, which is compatible with the connection. When the effective divisor is reduced, a stability condition for the parabolic weights α was introduced in [START_REF] Inaba | Moduli of stable parabolic connections, Riemann-Hilbert correspondence and geometry of Painlevé equation of type VI[END_REF] and [START_REF] Inaba | Moduli of parabolic connections on curves and the Riemann-Hilbert correspondence[END_REF]. Now we fix generic α so that α-stable = α-semistable. The moduli spaces of α-stable parabolic connections were constructed in [START_REF] Inaba | Moduli of stable parabolic connections, Riemann-Hilbert correspondence and geometry of Painlevé equation of type VI[END_REF] and [START_REF] Inaba | Moduli of parabolic connections on curves and the Riemann-Hilbert correspondence[END_REF]. The cases where the underlying effective divisors are not necessary reduced were studied in [START_REF] Inaba | Moduli of unramified irregular singular parabolic connections on a smooth projective curve[END_REF]. In the cases, as in [START_REF] Inaba | Moduli of unramified irregular singular parabolic connections on a smooth projective curve[END_REF], we call the parabolic connections unramified irregular singular parabolic connections. A stability condition for the parabolic weights α was also introduced and the moduli spaces of α-stable unramified irregular singular parabolic connections were constructed in [START_REF] Inaba | Moduli of unramified irregular singular parabolic connections on a smooth projective curve[END_REF]. These moduli spaces play important roles to prove the geometric Painlevé property of the isomonodromic deformations of these connections ( [START_REF] Inaba | Moduli of stable parabolic connections, Riemann-Hilbert correspondence and geometry of Painlevé equation of type VI[END_REF], [START_REF] Inaba | Moduli of parabolic connections on curves and the Riemann-Hilbert correspondence[END_REF], and [START_REF] Inaba | Moduli of unramified irregular singular parabolic connections on a smooth projective curve[END_REF]). To study geometry of the moduli spaces of parabolic connections (or unramified irregular singular parabolic connections), the moduli spaces of parabolic bundles are useful. Indeed, we have a correspondence from (unramified irregular singular) parabolic connections to parabolic bundles by forgetting the connections. Remark that the corresponding parabolic bundles are not necessary α-stable if (unramified irregular singular) parabolic connections are α-stable. But, this correspondence induces maps from Zariski open subsets on the moduli spaces of connections to the moduli spaces of α-stable parabolic bundles. We may expect that these maps have nice properties (for example, these are affine bundles and Lagrangian fibrations).

In this paper, we impose that the ranks of underlying vector bundles are 2 and base curves are the projective line P 1 . That is, our parabolic bundle (E, l) is a pair where E is a rank 2 vector bundle over P 1 , l = {l i } 1≤i≤ν , and l i ⊂ E| ni[ti] is a free O ni[ti] -submodule of length n i for each i. So we will consider only special situations. But isomonodromic deformations in this situation are interesting. Indeed, these isomonodromic deformations correspond to • some irregular Garnier systems when deg(D) ≥ 5 and D is not reduced. On the other hand, the special situation was studied in the point of view of the geometric Langlands problem [START_REF] Arinkin | On the moduli of SL(2)-bundles with connections on P 1 \ {x 1[END_REF], [START_REF] Arinkin | An example of the Langlands correspondance for irregular rank two connections on P 1[END_REF], and [START_REF] Donagi | Parabolic Hecke eigensheaves[END_REF].

1.1. When the effective divisor is reduced. Our purpose is to extend some result on parabolic bundles in the cases where D is reduced. Now we recall some results on parabolic bundles with a reduced effective divisor D = [t 1 ] + • • • + [t ν ], which are concerned with the contents of the present paper. Let Λ = (λ + i , λ - i ) 1≤i≤ν be a tuple of complex numbers such that ν i=1 (λ + i + λ - i ) + d = 0. Here we impose ν i=1 λ ǫi i ∈ Z for any choice (ǫ i ) 1≤i≤ν ∈ {+, -}. We say Λ is generic if Λ satisfies this condition. We define a Λ-parabolic connection as a parabolic connection such that the two residual eigenvalues of the connection are λ + i and λ - i for each t i and the parabolic structure satisfies the compatibility condition with the connection and λ + i (for details, see the paragraph after the proof of Lemma 6 with n i = 1 for any i). We say a parabolic bundle is Λ-flat if it admits a Λ-parabolic connection. We have a criterium of Λ-flatness given by [3, Proposition 3]: (1.1) End(E, l) = C ⇐⇒ (E, l) is Λ-flat ⇐⇒ (E, l) is undecomposable.

1.2. Results. Now we describe results of the present paper. The effective divisor D is not necessary reduced. First we will discuss a counterpart of (1.1). Let Λ = (λ + , λ -) where λ ± ∈ Ω 1 (D)/Ω 1 . As in the reduced divisor cases, we impose that Λ satisfies some conditions. We define a Λ-unramified irregular singular parabolic connection as an unramified irregular singular parabolic connection such that the diagonalization of the principal parts of the connection at D is a diagonal matrix whose diagonal entries are λ + and λ -and the parabolic structure satisfies the compatibility condition with the connection and λ + (for details, see the paragraph after the proof of Lemma 6). We say a parabolic bundle is Λ-flat if it admits a Λ-unramified irregular singular parabolic connection. If deg(D) = 4 and deg(E) is odd, then

End(E, l) = C ⇐⇒ (E, l) is Λ-flat which is proved in [START_REF] Arinkin | An example of the Langlands correspondance for irregular rank two connections on P 1[END_REF]Proposition 4.8]. We can define undecomposability for parabolic bundles when the effective divisor D is not necessary reduced. If (E, l) is Λ-flat, then (E, l) is undecomposable. But the converse proposition is not true even when deg(D) = 4 and deg(E) is odd. So in addition to undecomposability, we assume the admissibleness for parabolic bundles, which is defined in Definition 8 below. Using some material of [START_REF] Arinkin | An example of the Langlands correspondance for irregular rank two connections on P 1[END_REF]Section 4] and ideas of [START_REF] Arinkin | On the moduli of SL(2)-bundles with connections on P 1 \ {x 1[END_REF]Proposition 3], we can show that (1.3) (E, l) is simple =⇒ (E, l) is Λ-flat =⇒ (E, l) is undecomposable and admissible. This is a counterpart of (1.1). Here we say (E, l) is simple if End(E, l) = C. Counterexamples of the converse of the first implication of (1.3) are described in Proposition 12 and Proposition 14 below. Counterexamples of the converse of the second implication of (1.3) also are described in Proposition 12 and Proposition 14 (see also Example 32). If deg(D) = 4 and deg(E) is odd, then we have (1.4) (E, l) is undecomposable and admissible =⇒ End(E, l) = C. The implications (1.3) and (1.4) are described in Proposition 9 below. These are the first results.

Second we will discuss a counterpart of (1.2). We can introduce a stability condition with respect to weights w ∈ (w i ) ∈ [0, 1] ν , which is defined in Section 4 below. Let Bun w (D, d) be the moduli space of w-semistable parabolic bundles. However Bun w (D, d) is not projective in general, since we impose that each l i is free in the definition of parabolic bundles. So, to construct a good compactification, we will define refined parabolic bundles, which are generalization of parabolic bundles. Let l i,• = {l i,k } 1≤k≤ni be a filtration ] modules such that the length of l i,k is k. We call l i,• = {l i,k } 1≤k≤ni a refined parabolic structure at t i . If l i,ni is free, then the notion of refined parabolic structures coincides with the notion of parabolic structures. Refined parabolic bundles are vector bundles with refined parabolic structures. For weights w = (w 1 , . . . , w ν ), w i = (w i,ni , . . . , w i,1 ) ∈ [0, 1] ni (1 ≤ i ≤ ν), where 0 ≤ w i,ni ≤ • • • ≤ w i,1 ≤ 1, we can define w-stability and w-semistability of a refined parabolic bundle (Definition 29 below). Remark 1. A compactification of the moduli space of α-stable Λ-parabolic connections is constructed in [START_REF] Inaba | Moduli of stable parabolic connections, Riemann-Hilbert correspondence and geometry of Painlevé equation of type VI[END_REF]Proposition 5.6]. This compactification is the moduli space of α-stable Λ-φ-parabolic connections. When D is reduced, deg(D) = 4 and rank(E) = 2, the compactification with the boundary divisor is isomorphic to the Okamoto-Painlevé pair of P V I ( [START_REF] Inaba | Moduli of stable parabolic connections, Riemann-Hilbert correspondence and geometry of Painlevé equation of type VI. II. Moduli spaces and arithmetic geometry[END_REF]). Here the notion of Okamoto-Painlevé pairs was introduced in [START_REF] Saito | Deformation of Okamoto-Painlevé pairs and Painlevé equations[END_REF]. In [START_REF] Miyazaki | On compactifications of moduli of unramified irregular singular connections and Okamoto-Painlevé pairs[END_REF], a compactification of the moduli space of α-stable Λ-unramified irregular singular parabolic connections was studied when D is not reduced, deg(D) = 4 and rank(E) = 2. However, the moduli space of α-stable Λ-φ-unramified irregular singular parabolic connections is projective but is not smooth. So, in [START_REF] Miyazaki | On compactifications of moduli of unramified irregular singular connections and Okamoto-Painlevé pairs[END_REF], Miyazaki introduced refined parabolic structures, which are same as our refined parabolic structures. By considering refined parabolic structures, we have smooth compactifications, which are isomorphic to the Okamoto-Painlevé pairs of P V , P IV , P III(D (1) 6 ) , and P II . Remark 2. The notion of the refined parabolic bundles is contained in the notion of the parabolic bundles in Yokogawa's papers [START_REF] Yokogawa | Compactification of moduli of parabolic sheaves and moduli of parabolic Higgs sheaves[END_REF] and [START_REF] Yokogawa | Infinitesimal deformation of parabolic Higgs sheaves[END_REF].

E| ni[ti] ⊃ l i,ni ⊃ l i,ni-1 ⊃ • • • ⊃ l i,1 ⊃ 0 of O ni[ti
We can define undecomposable refined parabolic bundles and admissible refined parabolic bundles, naturally. Remark 3. We have defined the Λ-flatness for (ordinary) parabolic bundles. On the other hand, we do not define the Λ-flatness for refined parabolic bundles, since unramified irregular singular parabolic connections have (ordinary) parabolic structures.

To give a counterpart of (1.2), the conditions undecomposableness and admissibleness are not sufficient. That is, there exist undecomposable and admissible refined parabolic bundles which is not w-stable for any weights w (Example 32 below).

Remark 4. Moreover, there exist examples of Λ-flat parabolic bundles, which are w-unstable for any weights w.

So we will define tame refined parabolic bundles in Definition 33. We will check that (E, l) is undecomposable and tame =⇒ (E, l) is admissible (described in Proposition 34 below). Then we have a counterpart of (1.2) as follows:

Theorem A (Theorem 1 and Corollary 2). Let (E, l) be a refined parabolic bundle of degree d over (P 1 , D). We have the following equivalence relation:

(E, l) is undecomposable and tame ⇐⇒ w-stable for a convenient choice of weights w.

Let (E, l) be a parabolic bundle of degree d over (P 1 , D). Then (E, l) is simple ⇐⇒ w-stable for a convenient choice of weights w.

Third, we will discuss elementary transformations of refined parabolic bundles. We will define elementary transformations of refined parabolic bundles in Section 5.3 and we will show some properties of the elementary transformations in Section 5.4. For example, we will discuss a relation between the elementary transformations and the stability condition (Proposition 43).

Fourth, we will describe the moduli space of w-stable refined parabolic bundles when deg(D) = 5, deg(E) = 1, and w is generic democratic weights. Let Bun w (D, d) be the moduli space of w-stable refined parabolic bundles of degree d. Here w-semistable = w-stable, since w is generic. In this case, the dimension of Bun w (D, d) is 2. Since deg(D) = 5, the effective divisor D is one of the following effective divisors:

D 2111 := 2[t 1 ] + [t 2 ] + [t 3 ] + [t 4 ], D 221 := 2[t 1 ] + 2[t 2 ] + [t 3 ], D 311 := 3[t 1 ] + [t 2 ] + [t 3 ], D 32 := 3[t 1 ] + 2[t 2 ], D 41 := 4[t 1 ] + [t 2 ], D 5 := 5[t 1 ]
. For these cases, by Theorem A, we can make a list of refined parabolic bundles which are w-stable for a convenient choice of weights w. This list is described in the appendix. We will define types of the refined parabolic bundles by some numerical data. This type is related to behavior of the stability index when weights are varied. By Proposition 47 below, we have that Bun w (D, 1) = P 2 when 1 5 < w < 1 3 . By this fact and the relation between the types and behavior of the stability index, we have the following theorem.

Theorem B (Section 6.1). Assume that deg(D) = 5, deg(E) = 1, and weight w = (w i ) i satisfies w i,1 = w i,2 = • • • = w i,ni = w for any i (0 < w < 1), that is, w are democratic weights.

(i) We have the following explicit descriptions of Bun w (D, 1):

-Bun w (D, 1) is empty when 0 < w < 1 5 ; -Bun w (D, 1) = P 2 when 1 5 < w < 1 3 ;

-Bun w (D, 1) is a weak del Pezzo surface of degree 4 when 1 3 < w < 3 5 ; -Bun w (D, 1) is a weak del Pezzo surface of degree 5 when 3 5 < w < 1. Let Π -2 (D) and Π ′ -2 (D) be the effective divisors consisting of all (-2)-curves on Bun w (D, 1) when 1 3 < w < 3 5 and 3 5 < w < 1, respectively. Then the configurations of Π -2 (D) and Π ′ -2 (D) are same, and the configuration is (ii) If (E, {l i,• } i ) be a refined parabolic bundle corresponding to a point on Π -2 (D) or Π ′ -2 (D), then there exists i (1 ≤ i ≤ ν) such that l i,ni is not free, that is, l i,• is not a parabolic structure for this i.

A 1 , 2A 1 , A 2 , A 2 + A 1 ,
(iii) If (E, {l i,• } i ) be a refined parabolic bundle corresponding to a point on Bun w (D, 1) ( 1 5 < w < 1 3 ), Bun w (D, 1) \ Π -2 (D) ( 1 3 < w < 3 5 ), or Bun w (D, 1) \ Π ′ -2 (D) ( 3 5 < w < 1)
, then l i,ni is free for any i, and (E, {l i,ni } i ) is a simple parabolic bundle. In particular, (E, {l i,ni } i ) is a Λ-flat for generic Λ. These weak del Pezzo surfaces are listed in [START_REF] Dolgachev | Classical algebraic geometry. A modern view[END_REF]Section 8.6.3]. Descriptions of the corresponding isomonodromic deformations are given by Kimura [START_REF] Kimura | The degeneration of the two-dimensional Garnier system and the polynomial Hamiltonian structure[END_REF] (see also [START_REF] Kawakami | Toward a classification of four-dimensional Painlevé-type equations[END_REF]). The corresponding isomonodromic deformations are the following: the divisor Π -2 the isomonodromic deformations

D 2111 A 1 H(1, 1, 1, 2) D 221 2A 1 H(1, 2, 2) D 311 A 2 H(1, 1, 3) D 32 A 2 + A 1 H(2, 3) D 41 A 3 H(1, 4) D 5
A 4 H(5) Here the notations in the last column are in [20, pp.39-40].

Fifth, we will recover the geometry of the weak del Pezzo surfaces of degree 4 from the modular point of view. When weights w are the democratic weights with w = 1 2 , the moduli spaces of refined parabolic bundles are weak del Pezzo surfaces of degree 4. We may reinterpret the negative curves on these surfaces as the loci of special refined parabolic bundles. Since weights w are the democratic weights with w = 1 2 , we may show that the stability of refined parabolic bundles is preserved by applying the elementary transformations (Section 6.2). So some elementary transformations induce automorphisms on the moduli space Bun w (D, d) with w = 1 2 . We will reconstruct automorphisms of the weak del Pezzo surfaces of degree 4 via these elementary transformations of refined parabolic bundles.

The organization of this paper is as follows. In Section 2, we will define parabolic bundles over (P 1 , D), where D is not necessary reduced. Next, in Section 2.1, we will show the implications (1.3) and (1.4), and will give a flatness criterium. In Section 2.2, we will give examples of the flatness criterium. In Section 3, we will define refined parabolic bundles and will discuss combinatorial structure of refined parabolic structures at a multiple point. This structure is related to Young tableaus. In Section 4, we will define a stability condition of refined parabolic bundles, and we will show Theorem A. In Section 5.3, we will define elementary transformations of refined parabolic bundles. In Section 5.4, we will show some properties of the elementary transformations of refined parabolic bundles. In Section 6.1, we will describe the moduli spaces when deg(D) = 5, deg(E) = 1, and weights w are democratic weights. In Section 6.3, we will recover the geometry of the weak del Pezzo surfaces of degree 4 from the modular point of view. In the appendix, we will give a list of special refined parabolic bundles, which appear when deg(D) = 5 and deg(E) = 1.

Acknowledgments. The authors would like to warmly thank Professor Roman Fedorov for illuminating discussion helping us to understand the flatness criterium in Proposition 9. They also thank the referee for useful remarks.

Parabolic bundles

Let D be an effective divisor on P 1 , defined by

D = ν i=1 n i [t i ], deg(D) = ν i=1 n i =: n. Take a generator f i of the maximal ideal of O P 1 ,ti . Set O ni[ti] := O P 1 ,ti /(f ni i ).
For a vector bundle E on P 1 , we denote by

E| ni[ti] the tensor product E ⊗ O ni[ti] . Let I = {1, 2, . . . , ν} be the set of indexes of D. Take Λ = (λ + , λ -), with λ ± ∈ Ω 1 (D)/Ω 1 . Set λ ± i := λ ± | ni[ti] ∈ Ω 1 (D)/Ω 1 ⊗ O ni[ti]
for each i ∈ I. We assume (as in [4, Section 2.1]):

(a) the order of pole of

λ + i -λ - i is precisely n i , (b) i Res ti (λ + i + λ - i ) = d ∈ Z, (c) i Res ti λ ± i ∈ Z whatever are the signs ±, (d) if n i = 1 then Res ti λ + i -Res ti λ - i ∈ Z. The relation in (b) is called Fuchs' relation and the condition (d) is called non resonant condition.
Let E be a rank 2 vector bundle on P 1 and ∇ : E → E ⊗ Ω 1 (D) be a meromorphic connection with polar divisor D.

Definition 5. We say (E, ∇) is Λ-connection if its principal part at each pole n i [t i ] is given by d + λ + i 0 0 λ - i : E| ni[ti] -→ E| ni[ti] ⊗ Ω 1 (D)/Ω 1
up to holomorphic gauge transformation. We call Λ the formal (or spectral) data of the meromorphic connection. Proof. If L ⊂ E is a ∇-invariant line bundle, then ∇| L : L → L ⊗ Ω 1 (D) is a connection with principal parts λ ± i at each t i . Therefore, Fuchs' relation for ∇| L yields a contradiction with generic assumption (c) on residual eigenvalues.

If n i ≥ 2,
Let (E, ∇) be a Λ-connection. By the condition of Λ, there is a unique free O ni[ti] -submodule

l i ⊂ E| ni[ti]
of length n i for each i by the compatible with ∇ in the following sense: ∇(s)λ + i s is regular at t i for any i and every section s of E in a neighborhood of

t i such that s| ni[ti] ∈ l i . Let l = {l i } i∈I be the set of the free O ni[ti] -submodule l i ⊂ E| ni[ti]
of length n i given by a Λ-connection for each i. We call the triple (E, ∇, l) unramified irregular singular Λ-parabolic connection (see [START_REF] Inaba | Moduli of unramified irregular singular parabolic connections on a smooth projective curve[END_REF]Definition 2.1]). After fixing convenient weights α, the moduli space of α-stable unramified irregular singular Λ-parabolic connections of rank 2 on (P 1 , D) forms a smooth quasi-projective scheme of dimension 2(n -3) (see [START_REF] Inaba | Moduli of unramified irregular singular parabolic connections on a smooth projective curve[END_REF]Section 2]). In our case, by Proposition 6, unramified irregular singular Λ-parabolic connections induced by Λ-connections are irreducible. So these unramified irregular singular Λ-parabolic connections satisfy the stability condition in [START_REF] Inaba | Moduli of unramified irregular singular parabolic connections on a smooth projective curve[END_REF]Definition 2.2]. So we identify α-stable unramified irregular singular Λ-parabolic connections with Λ-connections. Let Con Λ (D) be the moduli space of Λ-connections. In this paper, we are concerned with forgetting the connections from unramified irregular singular Λ-parabolic connections. Definition 7. We say (E, l) (where l = {l i } i∈I ) is a parabolic bundle of rank 2 and of degree d over (P 1 , D) if E is a rank 2 vector bundle on P 1 with deg(E) = d and l i ⊂ E| ni[ti] is a free O ni[ti] -submodule of length n i for each i. We call l i a parabolic structure at t i .

2.1. Flatness criterium. We say a parabolic bundle (E, l) is Λ-flat if there exists a connection on E compatible with parabolic structure l and formal data Λ. We say that a parabolic bundle

(E, {l i } i∈I ) is decomposable if there exists a decomposition E = L 1 ⊕ L 2 such that l i = l (1) i or l i = l (2) i
for any i ∈ I, where we set l Proposition 3] and [START_REF] Loray | Lagrangian fibrations in duality on moduli spaces of rank 2 logarithmic connections over the projective line[END_REF]Proposition 3.1]). The identification of these three conditions is not necessarily true for any curves. So for reduced divisor cases, the target of the forgetting map from Con Λ (D) is the moduli space of undecomposable parabolic bundles. Here Con Λ (D) is the moduli space of Λ-connections. For n = 4 (that is, D = t 1 + • • • + t 4 ), the moduli space of rank 2 undecomposable parabolic bundles with degree 0 (or 1) can be obtained by glueing two copies of P 1 outside t 1 , . . . , t 4 (see [START_REF] Arinkin | An example of the Langlands correspondance for irregular rank two connections on P 1[END_REF]Section 2.3]). For n = 5 (that is, D = t 1 + • • • + t 5 ), Loray-Saito have provided a precise description of the moduli space of rank 2 undecomposable parabolic bundles with degree -1. This moduli space is closely related to the geometry of degree 4 del Pezzo surfaces. Now we consider cases where D is not necessary reduced. It is proved in [4, Proposition 4.8] that, if n = 4 (Painlevé case) and deg(E) is odd, then

(1) i := l i ∩ (L 1 | ni[ti] ) and l (2) i := l i ∩ (L 2 | ni[ti] ). We say that (E, {l i } i∈I ) is undecomposable if (E, {l i } i∈I ) is not decomposable. If D is a reduced divisor, then End(E, l) = C ⇐⇒ (E, l) is Λ-flat ⇐⇒ (E, l) is undecomposable (see [3,
End(E, l) = C ⇐⇒ (E, l) is Λ-flat.
We will check the following claim: If (E, l) is Λ-flat, then (E, l) is undecomposable. But the converse proposition is not true even when deg(D) = 4 and deg(E) is odd. So in addition to undecomposability, we assume the admissibleness defined below.

Remind that, on P 1 , any vector bundle is decomposable by Birkhoff:

E ≃ O(d 1 ) ⊕ O(d 2 ) with d 1 ≤ d 2 unique. The subbundle O(d 1 ) ֒→ E is not unique, but if d 1 < d 2 , then O(d 2 ) ֒→ E is unique. When d 1 = d 2 ,
the two subbundles of the decomposition can be arbitrarily choosen in a 1-parameter family. In any case, the (possible) factor L = O(d 2 ) of the decomposition is characterized by deg(E) ≤ 2 deg(L). Definition 8. We will say that (E, l) is admissible if the following condition is satisfied:

(2.1) ∀L ⊂ E, deg(E) ≤ 2 deg(L) =⇒ i∈I length(l i ∩ L| ni[ti] ) ≤ n + deg(E) -2 deg(L) -2.
Using some material of [4, Section 4] and ideas of [3, Proposition 3], we can prove: Proposition 9. Let Λ satisfying (a),(b),(c),(d) as above, and (E, l) a parabolic bundle of degree d. Then, we have implications (i) ⇒ (ii) ⇒ (iii) where:

(i) End(E, l) = C, (ii) (E, l) is Λ-flat, (iii) (E, l
) is undecomposable and admissible.

Moreover, for n = 4 and deg(E) odd, these are equivalent.

Proof. (ii) ⇒ (iii). Let ∇ be a connection on E compatible with the parabolic structure l and formal data Λ. Assume, by contradiction, that (E, l) = (L 1 , D 1 ) ⊕ (L 2 , D 2 ), meaning that the vector bundle splits as E = L 1 ⊕ L 2 for line bundles L i , and parabolic directions are distributed on L i over D i , i = 1, 2, where D = D 1 + D 2 with D 1 , D 2 having disjoint support. Then, with respect to that decomposition, the connection ∇ writes

∇ 1 θ 1,2 θ 2,1 ∇ 2
where θ i,j ∈ Hom(L j , L i ⊗ Ω 1 (D)) and ∇ i :

L i → L i ⊗ Ω 1 (D) a connection.
Then, the principal part at t i writes:

either λ + i * 0 λ - i , or λ - i 0 * λ + i .
Therefore, Fuchs' relation for L 1 (or L 2 ) yields a contradiction with generic assumption (c) on residual eigenvalues.

On the other hand, let L ⊂ E be an arbitrary proper subbundle of E. We consider the composition map

L ֒→ E ∇ → E ⊗ Ω 1 (D) ։ (E/L) ⊗ Ω 1 (D).
This map ϕ L is O-linear and it is not the zero map, otherwise L would be ∇-invariant, therefore contradicting Proposition 6. We deduce that

m := deg(E) -2 deg(L) + n -2 ≥ 0.
Moreover, m is the number of zeroes of ϕ L (counted with multiplicities). If

l i ∩ L| ni[ti] = 0, then l i ∩ L| ni[ti]
contributes for zeroes of ϕ L , and we therefore deduce

i∈I length(l i ∩ L| ni[ti] ) ≤ deg(E) -2 deg(L) + n -2
which implies, when applied to the case deg(E) ≤ 2 deg(L), that (E, l) is admissible.

(i) ⇒ (ii). Let (E, l) be as in (i). We can locally trivialize E| Uα ≡ C 2 with parabolic structure normalized as 1 0 , so that we can locally define a Λ-connection compatible with the parabolic structure on U α by

∇ α = d + λ + α 0 0 λ - α where λ ± α is a 1-form on U α with principal par λ ± | Uα . On intersections U α ∩ U β , the difference ∇ α -∇ β = θ αβ
is a section of the sheaf of parabolic Higgs fields Higgs(E, l) which is the subsheaf of End(E, l) ⊗ Ω 1 (D) whose principal parts are nilpotent. In above trivialization, we have

θ αβ = 0 * 0 0 + holomorphic.
This defines an element of {θ αβ } ∈ H 1 (P 1 , Higgs(E, l)) and the existence of a global Λ-connection ∇ compatible with the parabolic structure is equivalent to the vanishing of this cocycle:

∇ Uα = ∇ α -θ α with θ α -θ β = θ αβ .
In order to prove that {θ αβ } ≡ 0, we are going to use Serre duality, i.e. there is a perfect pairing

H 1 (P 1 , Higgs(E, l)) × H 0 (P 1 , End(E, l)) → H 1 (P 1 , Ω 1 ) ∼ → C ({θ αβ }, A) → {ω αβ := tr(θ αβ • A)}
where End(E, l) is the sheaf of parabolic endomorphisms. If {ω αβ } ≡ 0 for all A, then we can conclude that {θ αβ } ≡ 0 and therefore that there exists a global unramified irregular singular Λparabolic connection on (E, l). By assumption (i), each A takes the form c

θ αβ = (∇ α -∇ 0 ) -(∇ β -∇ 0 );
therefore, {ω αβ } can also be viewed as the meromorphic Čech coboundary

{ω αβ } = ∂ ω α := tr (∇ α -∇ 0 ) • A .
Then we have:

{ω αβ } ∼ 0 ⇐⇒ Res(ω α ) = 0
(note that residue does not depend on the choice of α as residue of ∇ α -∇ β is nilpotent). Indeed, the vanishing of residue is equivalent to the existence of a global meromorphic 1-form ω such that ωω α is holomorphic, i.e. such that {ω αβ } is a holomorphic Čech coboundary: (iii) ⇒ (i) in the case n = 4 and deg(E) odd. Let A be an endomorphism of the parabolic bundle (E, l). Since eigenvalues of A| Ex are holomorphic functions of x ∈ X, they are constant c 1 , c 2 ∈ C. If c 1 = c 2 , then the corresponding eigenspaces define a decomposition E = L 1 ⊕ L 2 and parabolic structure, since it is A-invariant, must be distributed in L 1 and L 2 ; this implies that (E, l) is decomposable, contradiction. We therefore have c 1 = c 2 =: c, and any endomorphism of the parabolic bundle takes the form A = c • Id E + N with N nilpotent. Clearly, c • Id E ∈ End(E, l), and (i) is equivalent to say that there is no non-zero nilpotent endomorphism in End(E, l).

{ω αβ } = ∂{ω α -ω} meaning that {ω αβ } ∼ 0 in H 1 (P 1 , Ω 1 ). If A = Id E ,
Assume, by contradiction, that N is a non-zero nilpotent element of End(E, l). Let L 0 = ker(N ) = image(N ). Since N is not trivial, it induces a non trivial morphism E/L 0 → L 0 . This implies that deg(E) -2 deg(L 0 ) ≤ 0. In particular, we have a decomposition E = L ⊕ L 0 with deg(L) ≤ deg(L 0 ) and, through that decomposition, the nilpotent endomorphism N writes:

N = 0 0 f 0 where f = f (x) is a polynomial of degree 2 deg(L 0 ) -deg(E). Locally at n i [t i ],
the parabolic structure is given, through the splitting and local coordinate z = xt i , by

z mi g(z) 1
where

m i = i∈I length(l i ∩ L 0 | ni[ti]
), and g(0) = 0. The action of N on the parabolic writes

0 0 f 0 • z mi g 1 = 0 z mi f g
so that N preserves the parabolic structure if, and only if, f (and therefore N ) vanish at t i at order n i -2m i . Indeed, we have

(2.2) det z mi g 0 1 z mi f g = z 2mi f g 2 = 0 mod z ni iff f = z ni-2mi f
and the induced action of N is by multiplication

0 0 f 0 • z mi g 1 = z mi f g • z mi g 1 mod z ni .
We promptly deduce that

(2.3) n -2 i∈I length(l i ∩ L 0 | ni[ti] ) ≤ deg(f ) ≤ 2 deg(L 0 ) -deg(E).
On the other hand, from (iii), (E, l) is admissible and we have that

i∈I length(l i ∩ L 0 | ni[ti] ) ≤ n + deg(E) -2 deg(L 0 ) -2.
We deduce

n + deg(E) -2 deg(L 0 ) 2 ≤ i∈I length(l i ∩ L 0 | ni[ti] ) ≤ n + deg(E) -2 deg(L 0 ) -2.
This implies, by comparing left and right hand-sides, that

n ≥ 2 deg(L 0 ) -deg(E) ≥0 +4
which is impossible when n = 4 and deg(E) odd. Therefore, N = 0 and End(E, l) = C.

The undecomposability and admissibleness are independent of Λ. So, when D is reduced or when n = 4 and deg(E) is odd, the Λ-flatness is independent of Λ. But, in general, the Λ-flatness depends on Λ. Indeed, we have the following proposition.

Proposition 10 ([4, Corollary 4.4]). Let Λ satisfying (a),(b),(c),(d) as above, and (E, l) a parabolic bundle of degree d. Assume (E, l) is undecomposable. Then (E, l) is Λ-flat if, and only if, for any nilpotent endomorphism N of (E, l), we have

(2.4) Res(N |l • (λ + -λ -)) = 0
where N |l denotes the action induced by N on the parabolic structure, and Res is the sum of residues.

Proof. We rephrase and continue the part [(iii) ⇒ (i)] of the proof of Proposition 9. The obstruction to the existence of a connection is defined by an element {∇ α -∇ β } ∈ H 1 (P 1 , Higgs(E, l)). By Serre duality, it is zero if, and only if Res(tr ∇ α -∇ 0 ) • A = 0 for all endomorphism A of (E, l); here ∇ 0 is a global meromorphic connection that we have fixed a priori. Since (E, l) is undecomposable, any endomorphism takes the form A = c • Id E + N with N nilpotent. We already know that the condition is satisfied for A = Id E because of Fuchs' relation and assumption (b) for Λ; by linearity, it is enough to consider A = N nilpotent. A non zero nilpotent endomorphism takes the form

N = 0 0 f 0 in a decomposition E = L ⊕ L 0 with deg(L) ≤ deg(L 0 ).
In fact, we can choose ∇ 0 such that it has one single pole with residue

k 1 0 0 k 2 dz z so that the contribution of ∇ 0 in Res(tr ∇ α -∇ 0 ) • A is zero. Let us compute the contribution of ∇ α .
Let the parabolic structure be defined by g 1 (e.g. in local coordinate near a pole). Since it is N -invariant, we have that f g 2 restricts to zero on D (or n i [t i ]), that we note f g 2 ∼ 0, and

0 0 f 0 • g 1 = 0 f g ∼ f g g 1
so that the induced action by (nilpotent) multiplication N | D is the multiplication by f g. For instance, if g vanishes at t i with multiplicity m i , then f vanishes with multiplicity n i -2m i , and

N | D = f g vanishes with multiplicity n i -m i . By gauge transformation 1 -g 0 1 0 0 f 0 1 g 0 1 ∼ -f g 0 f f g , we can assume l = 0 1 and therefore ∇ α = λ -0 * λ + . Then, we finally get (∇ α -∇ 0 ) • N = -f gλ - 0 * f gλ + so that Res(tr ∇ α -∇ 0 ) • N = Res(f g(λ + -λ -)) = Res(N |l • (λ + -λ -))
must be zero for any nilpotent endomorphism N of (E, l).

2.2.

Examples. We will give counterexamples of the implication [(ii) ⇒ (i)] and the implication [(iii) ⇒ (ii)], which are converses of the implications in Proposition 9. To give counterexamples, we will use the criterium of Proposition 10. First, we consider an example of this criterium. 1) and the parabolic structure l intersect the destabilizing line bundle O(1) only at [0] + [START_REF] Araujo | On automorphisms of moduli spaces of parabolic vector bundles[END_REF] (without multiplicity). In other words, the parabolic structure is given by

Example 11. Assume n = 5 with divisor 2[0]+2[1]+[∞], E = O⊕O(
c 0 x 1 mod x 2 , c 1 (x -1) 1 mod (x -1) 2 , and 1 0 at ∞.
Then (E, l) is undecomposable provided that (c 0 , c 1 ) = (0, 0). On the other hand, there is (up to homothecy) one non trivial nilpotent endomorphism

N = 0 0 1 0
that preserves the parabolic structure. It induces the multiplication by c 0 x and c 1 (x-1) on parabolic directions over x = 0 and x = 1 respectively. Up to bundle isomorphism, we see that only the projective variable [c 0 : c 1 ] ∈ P 1 makes sense. Criterium of Proposition 10 writes

Res(c 0 x(λ + 0 -λ - 0 ) + c 1 (x -1)(λ + 1 -λ - 1
)) = 0. Recall that dominant terms of λ +λ -are non zero at x = 0 and x = 1, given say by a 0 • Assume that (E, l) is undecomposable and admissible.

If (E, l) is not simple, then E = L ⊗ (O ⊕ O) and (i) D = 2[t 1 ] + 2[t 2
] and parabolic structure writes

c 1 (x -t 1 ) 1 and c 2 (x -t 2 ) 1
where

[c 1 : c 2 ] ∈ P 1 ; (ii) D = 4[t 1
] and parabolic structure writes 3 1

c 1 (x -t) 2 + c 2 (x -t)
where [c 1 : c 2 ] ∈ P 1 . • The only Λ-flat parabolic bundles that are not simple are the following ones: E = L⊗(O⊕O)

and

-D = 2[t 1 ] + 2[t 2
] and parabolic structure writes as in (i) where [c 1 : c 2 ] ∈ P 1 is a point determined by Λ; -D = 4[t 1 ] and parabolic structure writes as in (ii) where [c 1 :

c 2 ] ∈ P 1 is a point determined by Λ. Proof. Setting k = max L (2 deg(L)-deg(E)), L 0 be maximizing and m = i∈I length(l i ∩L 0 | ni[ti] ).
Then, end of proof of Proposition 9 yields

n -k 2 ≤ m ≤ n -k -2 so that in particular k + 4 ≤ n. It follows, when n = 4 that k = 0 and m = 2. Since k = 0, then E = L ⊗ (O ⊕ O).
On the other hand, assuming (E, l) is not simple, we know that there is a non trivial nilpotent endomorphism which writes N = 0 0 1 0 and L 0 is the invariant subspace. We deduce that all parabolics intersect L 0 , and at each pole t i , we must have n i -2m i ≤ 0; otherwise, parabolic structure would not be invariant. Together with above inequalities, we get n i = 2m i at each t i , whence the only two possibilities of the statement.

In each case, writing down the criterium of Proposition 10, as in Example 11, we get the restriction on (c 1 : c 2 ).

Now we consider

. If (E, l) is not simple, then E = L ⊗ (O ⊕ O(1)) and (i) D = 2[t 1 ] + 2[t 2 ] + [t 3
] and parabolic structure writes

c 1 (x -t 1 ) 1 , c 2 (x -t 2 ) 1 and 1 0
where

[c 1 : c 2 ] ∈ P 1 satisfies c 1 c 2 = 0; (ii) D = 3[t 1 ] + 2[t 2
] and parabolic structure writes

c 1 (x -t 1 ) + c 2 (x -t 1 ) 2 1 and c 3 (x -t 2 ) 1
where

[c 1 : c 2 : c 3 ] ∈ P 2 satisfies c 1 c 3 = 0; (iii) D = 4[t 1 ] + [t 2
] and parabolic structure writes

c 1 (x -t 1 ) 2 + c 2 (x -t 1 ) 3 1 and 1 0
where 4 1

[c 1 : c 2 ] ∈ P 1 satisfies c 1 = 0; (iv) D = 5[t 1 ] and parabolic structure writes c 1 (x -t 1 ) 2 + c 2 (x -t 1 ) 3 + c 3 (x -t 1 )
where

[c 1 : c 2 : c 3 ] ∈ P 2 satisfies c 1 = 0. In particular, when D = 2t 1 + t 2 + t 3 + t 4 or D = 3t 1 + t 2 + t 3 , we have that End(E, l) = C ⇐⇒ (E, l) is Λ-flat ⇐⇒ (E, l) is undecomposable and admissible. Proof. Setting k = max L (2 deg(L)-deg(E)), L 0 be maximizing and m = i∈I length(l i ∩L 0 | ni[ti] ).
Since (E, l) is not simple, we have k + 4 ≤ n as in the proof of the previous proposition. By this inequality, we have k = 0 or k = 1. Since E has odd degree, k = 0. So we have k = 1. This means that E = L ⊗ (O ⊕ O(1)). Since (E, l) is not simple, we know that there is a non trivial nilpotent endomorphism which writes

N = 0 0 f (x) 0 and L 0 is the invariant subspace. Here f (x) is a polynomial in x with deg(f ) ≤ 1. For one of indexes in I, m i satisfies a condition n i -2m i ≤ 1, and, for other indexes, m i satisfies a condition n i -2m i ≤ 0. In particular m i ≥ 2. Since (E, l) is admissible, we have m i ≤ 2.
Then for one of indexes in I, n i -2m i = 1, and, for other indexes, n i -2m i = 0. There exist the only four possibilities of the statement. Then, the only Λ-flat parabolic bundles that are not simple are the following ones: E = L⊗(O⊕O(1)) and

• D = 2[t 1 ] + 2[t 2 ] + [t 3
] and parabolic structure writes as in (i) of Lemma 13 where [c 1 :

c 2 ]
is a point determined by Λ; Proof. By Proposition 9, we have that (E, l) is undecomposable and admissible. Then, the parabolic structure write as in Lemma 13. In each case, writing down the criterium of Proposition 10, as in Example 11.

• D = 3[t 1 ] + 2[t 2 ]

Refined parabolic bundles

In the previous section, we have defined parabolic bundles of rank 2 and of degree d over (P 1 , D). In this section, we will consider a generalization of parabolic bundles, which will be called refined parabolic bundles.

3.1.

Cases where D is a reduced effective divisor. Here we assume that D is a reduced effective divisor. In [START_REF] Loray | Lagrangian fibrations in duality on moduli spaces of rank 2 logarithmic connections over the projective line[END_REF], the moduli space of undecomposable parabolic bundles of rank 2 and of degree d was studied. Since now we assume that D is reduced, this moduli space coincides with the moduli space of Λ-flat parabolic bundles of rank 2 and of degree d. We denote by Bun Λ (D, d) the moduli space of Λ-flat parabolic bundles with degree d. This moduli space is the image of the forgetting map:

Con Λ (D) -→ Bun Λ (D, d) (E, ∇) -→ (E, l).
Here Con Λ (D) is the moduli space of Λ-connections. This forgetting map is important for understanding the moduli space Con Λ (D). But the moduli space Bun Λ (D, d) is a non-separated scheme. To get a good moduli space, we have to impose a stability condition with respect to w = (w i ) ∈ [0, 1] n . We denoted by Bun w (D, d) the moduli space of w-semistable parabolic bundles of rank 2 and of degree d. The moduli space Bun w (D, d) is a normal irreducible projective variety. The open subset of w-stable parabolic bundles is smooth. For generic weights, we have that w-semistable = w-stable. So Bun w (D, d) is a smooth projective variety. An important fact is that (E, {l i } i∈I ) is stable for a convenient choice of weights w if, and only if, it is undecomposable (see [START_REF] Loray | Lagrangian fibrations in duality on moduli spaces of rank 2 logarithmic connections over the projective line[END_REF]Proposition 3.4]). So the moduli space Bun Λ (D, d) is covered by Bun w (D, d) for some w.

3.2.

Definition of refined parabolic bundles. We try to extend the story in Section 3.1 for cases where D are not necessary reduced. When D is not a reduced divisor, there are some problems:

(1) undecomposable parabolic bundles are not necessary Λ-flat and are not necessary w-stable for some weights w; (2) Bun w (D, d) is not necessary projective for generic weights w. (Remark that we impose that l i is free). Here the stability condition for parabolic bundles when D is not necessary reduced is defined in Section 4 below.

For the problem (1), in Section 4.1 and Section 4.2 below, we will discuss on a necessary and sufficient condition for the condition that there exist weights w such that a parabolic bundle is w-stable. The present section is concerned with the problem (2), that is, constructing a smooth compactification of Bun w (D, d). To take a good compactification, we consider a generalization of parabolic bundles as follows.

Definition 15. We say (E, {l i,• } i∈I ) is a refined parabolic bundle of rank 2 and of degree d if E is a rank 2 vector bundle on

P 1 with deg(E) = d and l i,• = {l i,k } 1≤k≤ni is a filtration E| ni[ti] ⊃ l i,ni ⊃ l i,ni-1 ⊃ • • • ⊃ l i,1 ⊃ 0 of O ni[ti]
-modules where the length of l i,k is k. We call this filtration a refined parabolic structure.

Let (E, {l i } i∈I ) be a parabolic bundle of rank 2 and of degree d. For (E, {l i } i∈I ), we can define a refined parabolic bundle of rank 2 and of degree d as follows: We define a filtration l i,• as

l i,• : E| ni[ti] ⊃ l i ⊃ f i • l i ⊃ f 2 i • l i ⊃ • • • ⊃ f ni-1 i • l i ⊃ 0
for each i. Here f i is a generator of the maximal ideal of O P 1 ,ti . Then we have a refined parabolic bundle (E,

{l i,• } i∈I ). Conversely if a refined parabolic structure E| ni[ti] ⊃ l i,ni ⊃ l i,ni-1 ⊃ • • • ⊃ l i,1
⊃ 0 where l i,ni is free, then the refined parabolic structure coincides with the refined parabolic structure induced by the parabolic structure l i,ni . So refined parabolic bundles are generalization of parabolic bundles. In Section 4 below, we will define a stability condition for refined parabolic bundles. First of all, in present section, we discuss refined parabolic structures at a multiple point.

We will describe a combinatorial structure on a refined parabolic structure at a multiple point.

3.3.

Refined parabolic structure at a multiple point. Let E be a rank 2 vector bundle on P 1 . We take a point t on P 1 and a positive integer n such that n > 1. First, we will define the type

of an O n[t] -submodule of E| n[t]
. We consider a filtration of sheaves

E ⊃ E(-[t]) ⊃ E(-2[t]) ⊃ • • • ⊃ E(-(n -1)[t]) ⊃ E(-n[t]). Note that E| n[t] = E/E(-n[t]). Set V j := E(-(n -j + 1)[t])/E(-n[t]) for j = 1, 2, . . . , n + 1.
Then we have a filtration

E| n[t] = V n+1 ⊃ V n ⊃ V n-1 ⊃ • • • ⊃ V 2 ⊃ V 1 = 0 of O n[t] -modules.
Definition 16. We fix a positive integer k where

1 ≤ k ≤ n. Let l be an O n[t] -submodule of E| n[t]
with length(l) = k. We define a tuple of integers λ := (λ n , λ n-1 , . . . , λ 1 ) by

λ j := length ((V j+1 ∩ l)/(V j ∩ l))
for j = 1, 2, . . . , n. We call the tuple λ the type of the O n[t] -submodule l with length(l) = k.

We have the equality n j=1 λ j = k. The inequalities λ j ≥ 0 and

λ j ≤ length(V j+1 /V j ) = 2
follow for j = 1, 2, . . . , n. Let f be a generator of the maximal ideal of O P 1 ,t . The map from V j+1 to V j defined by multiply by f induces an injective map

(V j+1 ∩ l)/(V j ∩ l)
multiply by f / / (V j ∩ l)/(V j-1 ∩ l) for j = 2, 3, . . . n. Then we have the following inequality

λ j = length((V j+1 ∩ l)/(V j ∩ l)) ≤ length((V j ∩ l)/(V j-1 ∩ l)) = λ j-1
for j = 2, 3, . . . n. For a tuple of integers λ = (λ n , λ n-1 , . . . , λ 1 ), we set

(3.2) 0 ≤ λ n ≤ λ n-1 ≤ • • • ≤ λ 1 ≤ 2 and n j=1 λ j = k, then there exists an O n[t] -submodule of E| n[t]
with length(l) = k whose type is λ.

Proof. We define v and v ′ as

v = f n-a1(λ)-a2(λ) v 1 v 2 ∈ O n[t] ⊕ O n[t] v ′ = f n-a2(λ) v ′ 1 v ′ 2 ∈ O n[t] ⊕ O n[t]
so that the image of v under the quotient map

(f n-a1(λ)-a2(λ) ) ⊕2 -→ ((f n-a1(λ)-a2(λ) )/(f n-a1(λ)-a2(λ)+1 )) ⊕2 ∼ = C 2 and the image of v ′ under the quotient map (f n-a2(λ) ) ⊕2 -→ ((f n-a2(λ) )/(f n-a2(λ)+1 )) ⊕2 ∼ = C 2 are linearly independent. Let l be the O n[t] -submodule of O n[t] ⊕ O n[t] generated by v and v ′ . The elements f a1(λ) v and v ′ are linearly independent in ((f n-a2(λ) ) ⊕2 ∩l)/((f n-a2(λ)+1 ) ⊕2 ∩l). In other words, length ((f n-a2(λ) ) ⊕2 ∩ l)/((f n-a2(λ)+1 ) ⊕2 ∩ l) = 2. If we take an isomorphism E| n[t] ∼ = O n[t] ⊕ O n[t]
, then l induces a refined parabolic structure on E| n[t] via this isomorphism. The type of this refined parabolic structure is λ.

Let λ = (λ n , λ n-1 , . . . , λ 1 ) be the type of an O n[t] -submodule of E| n[t]
. The type λ is a partition of the positive integer k: Proof. Let a 1 (λ) and a 2 (λ) be the integers defined in (3.1). We take v such that the image of v under the map

k = n 0 + • • • + 0 + 1 + • • • + 1 a1(λ) + 2 + • • • + 2
V a1(λ)+a2(λ)+1 → V a1(λ)+a2(λ)+1 /V a1(λ)+a2(λ) is nonzero.
Now we assume that the number of columns of the corresponding Young diagram is 1. That is, a 2 (λ) = 0. We consider an element v ′ ∈ l. There exists an integer

n v ′ such that v ′ ∈ V n v ′ +1 and the image of v ′ under the map V n v ′ +1 → V n v ′ +1 /V n v ′ is nonzero. We have a 1 (λ) ≥ n v ′ . The element f a1(λ)-n v ′ v is in V n v ′ +1 . Since λ n v ′ = 1, the images of f a1(λ)-n v ′ v and v ′ under the map V n v ′ +1 → V n v ′ +1 /V n v ′ are linearly dependent. Then there exist complex numbers α 1 and α 2 such that α 1 f a1(λ)-n v ′ v + α 2 v ′ ∈ V n v ′ . Moreover, the images of f a1(λ)-n v ′ +1 v and α 1 f a1(λ)-n v ′ v + α 2 v ′ under the map V n v ′ → V n v ′ /V n v ′ -1 are linearly dependent, since λ n v ′ -1 = 1.
By repeating this argument, we may write the element v ′ ∈ l as a linear combination of v, f v, f 2 v, and so on. Then l is generated by v.

Next we assume that the number of columns of the corresponding Young diagram is 2. That is, a 2 (λ) > 0. There exists an element w such that the image of w under the map V a2(λ)+1 → V a2(λ)+1 /V a2(λ) is nonzero and the images of f a1(λ) v and w under the map V a2(λ)+1 → V a2(λ)+1 /V n2 are linearly independent. We may show that l is generated by v and w by using the argument as in the case a 2 (λ) = 0.

Let v be an element of E| n[t] . We denote by v O n[t] the O n[t] -submodule of E| n[t] generated by v. If we take another generator v ′ of v O n[t] , then there is an element F ∈ O n[t] such that v = F •v ′ . We assume that l is generated by elements v 1 , v 2 of E| n[t] , where length( v 1 O n[t] ) ≥ length( v 2 O n[t] ). If we take another generator v ′ 1 and v ′ 2 of l, where length( v ′ 1 O n[t] ) ≥ length( v ′ 2 O n[t] ), then there is an element F 1 , F 2 , G 1 , G 2 ∈ O n[t] such that v 1 = F 1 • v ′ 1 + F 2 • v ′ 2 and v 2 = G 1 • f d • v ′ 1 + G 2 • v ′ 2 , where d = length( v ′ 1 ) -length( v ′ 2 ). Let l • = {l k } 1≤k≤n be a filtration E| n[t] ⊃ l n ⊃ l n-1 ⊃ • • • ⊃ l 1 ⊃ 0 of O n[t] -submodules of E| n[t] where the length of l k is k. Let λ (k) = (λ (k) n , λ (k) n-1 , . . . , λ (k)
1 ) be the type of the O n[t] -module l k for k = 1, 2, . . . , n. Then we have a sequence of Young diagrams (1) , where the skew diagram λ (k) /λ (k-1) consists of one box.

λ (n) ⊃ λ (n-1) ⊃ • • • ⊃ λ
Definition 19. Let l • = {l k } 1≤k≤n be a filtration E| n[t] ⊃ l n ⊃ l n-1 ⊃ • • • ⊃ l 1 ⊃ 0 (length(l k ) = k for k = 1, 2, . . . , n) of O n[t]
-modules. We denote by T l• the corresponding standard Young tableau to the sequence of Young diagrams as above. We call T l• the standard tableau of l • .

3.4.

Parameter space of refined parabolic structures at a multiple point. Next, we will consider families of refined parabolic structures at a multiple point and their parameter spaces.

Here, when we consider families of refined parabolic structures, we fix the first submodule l n of refined parabolic structures. We will construct parameter spaces of families of refined parabolic structures. We will see that there exists a correspondence between their irreducible components and standard Young tableaus whose shapes are the type of l n .

Let λ = (λ n , λ n-1 , . . . , λ 1 ) be a Young diagram which satisfies the condition (3.2). Let a 1 (λ) and a 2 (λ) be the integers defined in (3.1). We may describe the Young diagram λ as (1 a1(λ) , 2 a2(λ) ). Let l be an O n[t] -submodule of E| n[t] with type λ. First we consider a parameter space of O n[t]submodules l ′ of l with length(l/l ′ ) = 1. We consider the case a 2 (λ) = 0. In this case, l is generated by an element v ∈ E| n[t] (see Lemma 18)

. An O n[t] -submodule l ′ of l with length(l/l ′ ) = 1 is unique: l ′ = f • l.
Here f is a generator of the maximal ideal of O P 1 ,t . For the case a 2 (λ) = 0, we have the following lemma.

Lemma 20. Assume that a 2 (λ) = 0. By taking generators of l, we may construct a family of O n[t] -submodules l ′ of l with length(l/l ′ ) = 1 parametrized by P 1 such that,

• when a 1 (λ) = 0:

-the type of the submodule l ′ corresponding to [0 : 1] is (1 a1(λ)-1 , 2 a2(λ) ), and -the type of the submodule l ′ corresponding to [α 1 : α 2 ], where α 1 = 0, is (1 a1(λ)+1 , 2 a2(λ)-1 ); • when a 1 (λ) = 0:

-the type of the submodule l ′ corresponding to any [α 1 :

α 2 ] ∈ P 1 is (1 a1(λ)+1 , 2 a2(λ)-1 ).
This family gives a bijection

P 1 -→ {l ′ | l ′ is an O n[t] -submodule of l such that length(l/l ′ ) = 1}.
Proof. We take generators v 1 and v 2 of l such that v 1 is nonzero in V a1(λ)+a2(λ)+1 /V a1(λ)+a2(λ) .

Let [α 1 : α 2 ] be a point of P 1 . We set

v ′ 0 := α 1 v 1 + α 2 v 2 , v ′ 1 = f v 1 , and v ′ 2 := f v 2 . Let l ′ be an O n[t] -submodule of l generated by v ′ 0 , v ′ 1 , and v ′ 2 .
We may check that length(l/l ′ ) = 1. Then we have an O n[t] -submodule of l corresponding to the point [α 1 : α 2 ] ∈ P 1 . When a 1 (λ) = 0, the type of the submodule l ′ corresponding to [0 : 1] is (1 a1(λ)-1 , 2 a2(λ) ), and the type of the submodule l ′ corresponding to [1 :

α 2 /α 1 ] is (1 a1(λ)+1 , 2 a2(λ)-1 ). When a 1 (λ) = 0, the type of the submodule l ′ corresponding to [α 1 : α 2 ] is (1 a1(λ)+1 , 2 a2(λ)-1
). So we have a map from P 1 to the set of O n[t] -submodules l ′ of l with length(l/l ′ ) = 1.

We will construct the inverse map as follows. Let l ′ be an O n[t] -submodule of l and length(l/l ′ ) = 1. Let λ ′ be the type of l ′ . First we assume that a 2 (λ) > a 2 (λ ′ ). We have a 1 (λ ′ ) = a 1 (λ) + 1 and a 2 (λ ′ ) = a 2 (λ) -1. In this case, we may take an element w of l ′ such that w is nonzero in V a1(λ)+a2(λ)+1 /V a1(λ)+a2(λ) . Since elements of l ′ are also elements of l, there exist complex numbers α 1 and α 2 such that w -

α 1 v 1 -α 2 v 2 ∈ f v 1 , f v 2 O n[t]
. Since (α 1 , α 2 ) = (0, 0), we have the ratio [α 1 : α 2 ] ∈ P 1 by w. But we may check that the ratio [α 1 : α 2 ] ∈ P 1 is unique for the choice of such an element w. Then the ratio [α 1 : α 2 ] ∈ P 1 corresponds to l ′ . Second we assume that a 2 (λ) = a 2 (λ ′ ). We have a 1 (λ ′ ) = a 1 (λ) -1. In particular, a 1 (λ) = 0. In this case, we may take an element w of l ′ such that w is nonzero in V a1(λ)+a2(λ) /V a1(λ)+a2(λ)-1 . We may check that

w ∈ f v 1 , v 2 O n[t] . Since length(l/l ′ ) = 1, there exists a nonzero complex number α such that w -αv 2 ∈ f v 1 , f v 2 O n[t]
. So we have the ratio [0 : α] ∈ P 1 by w. But we may check that the ratio [0 : α] ∈ P 1 is unique for the choice of such an element w. Then the ratio [0 : α] ∈ P 1 corresponds to l ′ . By these arguments, we have the inverse map.

Let λ (n) = (1 a1 , 2 a2 ) be a Young diagram and l n be an O n[t] -submodule of E| n[t] with type λ (n) . Before we consider families and parameter spaces of refined parabolic structures, we discuss families and parameter spaces of filtrations

l n ⊃ l n-1 ⊃ • • • ⊃ l n-m+1 (2 ≤ m ≤ n) with length(l n-k+1 /l n-k ) = 1 (k = 1, 2, . . . , m -1). If m = n,
then this filtration is a refined parabolic structure. We consider examples with m = 3. We will consider the following four sequences of Young diagrams:

T : λ (n) = (1 a1 , 2 a2 ) ⊃ (1 a1-1 , 2 a2 ) ⊃ (1 a1-2 , 2 a2 ) T ′ : λ (n) = (1 a1 , 2 a2 ) ⊃ (1 a1+1 , 2 a2-1 ) ⊃ (1 a1 , 2 a2-1 ) T ′′ : λ (n) = (1 a1 , 2 a2 ) ⊃ (1 a1-1 , 2 a2 ) ⊃ (1 a1 , 2 a2-1 ) T ′′′ : λ (n) = (1 a1 , 2 a2 ) ⊃ (1 a1+1 , 2 a2-1 ) ⊃ (1 a1+2 , 2 a2-2 ).
Here we assume that • a 1 ≥ 2 when we consider the sequence T , • a 2 ≥ 1 when we consider the sequences T ′ , • a 1 ≥ 1 and a 2 ≥ 1 when we consider the sequences T ′′ , and • a 2 ≥ 2 when we consider the sequence T ′′′ . We will construct families of filtrations with fixed l n corresponding to these sequences of Young diagrams. Here for a sequence of Young diagrams, the "corresponding" means that filtrations parametrized by a Zariski open subset of the parameter space have this sequence of Young diagrams. We take generators v 1 and v 2 of l n such that length

v 1 O n[t] = a 1 +a 2 and length v 2 O n[t] = a 2 .
Example 21. First we will construct a family corresponding to T . This is the easiest case. Filtrations with T are unique. This filtration is

l n = v 1 , v 2 O n[t] ⊃ f v 1 , v 2 O n[t] ⊃ f 2 v 1 , v 2 O n[t] .
So the parameter space corresponding to T is a point.

Example 22. Second we will construct a family corresponding to T ′ as follows. We define a family of filtrations with fixed l n :

l n-1 = α 1 v 1 + α 2 v 2 , f v 1 , f v 2 O n[t] l n-2 = α 1 f v 1 + α 2 f v 2 , f v 1 , f v 2 O n[t] = f v 1 , f v 2 O n[t] ,
which is parametrized by [α 1 : α 2 ] ∈ P 1 . We denote by C ln,T ′ this parameter space P 1 . If α 1 = 0, the sequences of types of the parametrized filtrations are T ′ . So filtrations parametrized by the affine line A 1 have the sequence T ′ . If α 1 = 0, the sequence of types of the parametrized filtration is T ′′ . So we have the parameter space C ln,T ′ corresponding to T ′ .

Example 23. Third we will construct a family corresponding to T ′′ as follows. We define a family of filtrations with fixed l n :

l ′ n-1 = f v 1 , v 2 O n[t] and l ′ n-2 = α ′ 1 f v 1 + α ′ 2 v 2 , f 2 v 1 , f v 2 O n[t] ,
which is parametrized by [α ′ 1 : α ′ 2 ] ∈ P 1 . So we have the parameter space C ln,T ′′ corresponding to T ′′ . This parameter space C ln,T ′′ is P 1 .

We may check that [0 : 1] ∈ C ln,T ′ and [1 : 0] ∈ C ln,T ′′ parametrize the same filtration:

l ′ n-1 = f v 1 , v 2 O n[t] and l ′ n-2 = f v 1 , f v 2 O n[t]
. So the parameter spaces C ln,T ′ and C ln,T ′′ are the projective lines and the intersection C ln,T ′ ∩ C ln,T ′′ is a point.

Proposition 24. Let F 2 → P 1 be the Hirzebruch surface of degree two. We denote by s 0 and p ∞ the 0-section and the fiber of F 2 → P 1 at ∞, respectively. We can construct a family of filtrations with fixed l n corresponding to T ′′′ . More precisely, by this family, s 0 is identified with C ln,T ′ , p ∞ is identified with C ln,T ′′ , and

F 2 \ (s 0 ∪ p ∞ ) is identified with    l n ⊃ l n-1 ⊃ l n-2 l n-k is an O n[t] -submodule of l n-k+1 such that length(l n-k+1 /l n-k ) = 1 (k = 1, 2) and the sequence of Young diagram is T ′′′    . Proof. Set U 0 := {[1 : x 1 ] | x 1 ∈ C} ⊂ P 1 and U ∞ := {[y 1 : 1] | y 1 ∈ C} ⊂ P 1 .
We consider F 2 as the gluing of U 0 × P 1 and U ∞ × P 1 as follows:

(U 0 × P 1 )| U0∩U∞ -→ (U ∞ × P 1 )| U0∩U∞ ([1 : x 1 ], [α 1,3 : α 2,3 ]) -→ ([y 1 : 1], [β 1,3 : β 2,3 ])
where y 1 = 1/x 1 , β 1,3 = α 1,3 , and β 2,3 = -x -2 1 α 2,3 . We will construct a family of filtrations l n ⊃ l n-1 ⊃ l n-2 with length(l n-k+1 /l n-k ) = 1 (k = 1, 2) parametrized by F 2 as follows. We take

generators v 1 and v 2 of l n such that length v 1 O n[t] = a 1 + a 2 and length v 2 O n[t] = a 2 . For ([1 : x 1 ], [α 1,3 : α 2,3 ]) ∈ U 0 × P 1 , we define l n-1 and l n-2 as l n-1 = v 1 + x 1 v 2 , f v 2 l n-2 = α 1,3 (v 1 + x 1 v 2 ) + α 2,3 f v 2 , f v 1 + x 1 f v 2 , f 2 v 2 .
On the other hand, for ([

y 1 : 1], [β 1,3 : β 2,3 ]) ∈ U ∞ × P 1 we define l ′ n-1 and l ′ n-2 as l ′ n-1 = y 1 v 1 + v 2 , f v 1 l ′ n-2 = β 1,3 (y 1 v 1 + v 2 ) + β 2,3 f v 1 , y 1 f v 1 + f v 2 , f 2 v 1 . On U 0 ∩U ∞ , we put y 1 = 1/x 1 , β 1,3 = α 1,3 , and β 2,3 = -x -2 1 α 2,3 . Then we have that l n-1 | U0∩U∞ = l ′ n-1 | U0∩U∞ and l n-2 | U0∩U∞ = l ′ n-2 | U0∩U∞ .
So we obtain a family of filtrations parametrized by F 2 . Points on the Zariski open subset α 1,3 = 0 in U 0 × P 1 parametrize filtrations whose sequences of types are T ′′′ . So we say that the parameter space F 2 corresponds to T ′′′ . The curve C ln,T ′ defined above is embedded into F 2 as the 0-section. Here the 0-section is defined by α 1,3 = 0 over U 0 and β 1,3 = 0 over U ∞ . The curve C ln,T ′′ defined above is embedded into F 2 as the fiber of the point

y 1 = 0 under F 2 → P 1 .
We observe that the dimension of the parameter space corresponding to a sequence of Young diagrams λ

(n) ⊃ λ (n-1) ⊃ λ (n-2) is # k n -1 ≤ k ≤ n, a 2 (λ (k-1) ) = a 2 (λ (k) ) -1 .
Next we consider an example with m = 4. That is, we will construct a family of filtrations

l n ⊃ l n-1 ⊃ l n-2 ⊃ l n-3 with length(l n-k+1 /l n-k ) = 1 (k = 1, 2, 3).
Proposition 25. We can construct a P 1 -bundle over F 2 such that this bundle has a 0-section and we can construct a family corresponding to

T 4 : λ (n) = (1 a1 , 2 a2 ) ⊃ (1 a1+1 , 2 a2-1 ) ⊃ (1 a1+2 , 2 a2-2 ) ⊃ (1 a1+3 , 2 a2-3 )
over this P 1 -bundle. Here we assume that a 2 ≥ 3.

Proof. Let l n ⊃ ln-1 ⊃ ln-2 be the family of filtrations parametrized by F 2 defined in the case corresponding to T ′′′ . We set

U 1 := U 0 × U 0 , U 2 := U 0 × U ∞ , U 3 := U ∞ × U 0 , U 4 := U ∞ × U ∞ .
These products give an affine open covering of F 2 . We consider the restriction ln-2

| U i for 1 ≤ i ≤ 4.
Since we assume that a 2 ≥ 3, we may take two elements ṽi 1 and ṽi

2 of ln-2 | U i such that ln-2 | U i = ṽi 1 , ṽi 2 O n[t]
. For each U i , we define a family of filtrations l n ⊃ li n-1 ⊃ li n-2 ⊃ li n-3 (so m = 4) parametrized by U i × P 1 as follows. We set li n-1 := ln-1

| U i , li n-2 := ln-2 | U i , and (3.3) li n-3 := α i 1,4 ṽi 1 + α i 2,4 ṽi 2 , f ṽi 1 , f ṽi 2 O n[t]
.

where

[α i 1,4 : α i 2,4 ] ∈ P 1 .
Then we glue {U i × P 1 } 1≤i≤4 so that these families patch together. Then we have a P 1 -bundle over F 2 . We denote by F → F 2 has the 0-section. We may assume that length( ṽi

1 O n[t] ) ≥ length( ṽi 2 O n[t]
) for each i. Then this 0-section is defined by α i 1,4 = 0 in (3.3) for each i (1 ≤ i ≤ 4). That is, by the patching of the families, the local sections α i 1,4 = 0 over U i patch together.

To describe parameter spaces of refined parabolic structures l n ⊃ l n-1 ⊃ • • • ⊃ l 1 ⊃ 0, we consider sequences of maps, which are P 1 -bundles or identity maps as follows. For T 4 , we define a sequence of maps:

F (4) ln,T 4 p3 --→ F (3) ln,T 4 p2 --→ F (2) ln,T 4 p1 --→ F (1) ln,T 4 = {pt} so that F (2)
ln,T 4 is the projective line, p 2 is the Hirzebruch surface of degree two, and p 3 is the P 1bundle in Proposition 25. We identify F (4) ln,T 4 with the parameter space of this family corresponding to T 4 . Let

T 5 : λ = (1 a1 , 2 a2 ) ⊃ (1 a1+1 , 2 a2-1 ) ⊃ (1 a1+2 , 2 a2-2 ) ⊃ (1 a1+1 , 2 a2-2 ).
For T 5 , we define

F (4) ln,T 5 p3 --→ F (3) ln,T 5 p2 --→ F (2) ln,T 5 p1 --→ F (1) ln,T 5 = {pt} so that F (2)
ln,T 5 is the projective line, p 2 is the Hirzebruch surface of degree two, F

ln,T 5 = F (4) (3)
ln,T 5 , and p 3 is the identity map. We can construct a family corresponding to T 5 over the 0-section of the P 1 -bundle in Proposition 25. So we may identify F (4) ln,T 5 with the parameter space of this family corresponding to T 5 . Now we consider families and parameter spaces of refined parabolic structures.

Proposition 26. We set λ (n) = (1 a1 , 2 a2 ). We fix an O n[t] -submodule l n of E| n[t] with length(l n ) = n whose type is λ (n)
. Let T ln be the set of standard Young tableaus whose shapes are the Young diagram λ (n) . For T = (λ (n) , λ (n-1) , . . . , λ (1) ) ∈ T ln , we can construct a sequence of maps:

F (n) ln,T pn-1 ----→ F (n-1) ln,T pn-2 ----→ • • • p2 ---→ F (2) ln,T p1 ---→ F (1) ln,T = {pt} such that • F (k+1) ln,T = F (k)
ln,T and p k is the identity when a 2 (λ (n-k+1) ) = a 2 (λ (n-k) ) • p k is a P 1 -bundle when a 2 (λ (n-k+1) ) = a 2 (λ (n-k) ) + 1 • there exists a family of refined parabolic structures l•,T :

E| n[t] ⊃ l n ⊃ ln-1,T ⊃ • • • ⊃ l1,
C a2 -→ l • : l n ⊃ l n-1 ⊃ • • • ⊃ l 1 ⊃ 0 l • is a refined parabolic structure on n[t]
whose standard tableau is T .

Proof. We will construct a sequence of maps inductively. We assume that we have a sequence of maps

F (k) ln,T p k-1 ----→ F (k-1) ln,T p k-2 ----→ • • • p2 --→ F (2) ln,T p1 --→ F (1)
ln,T = {pt} and such a family l n ⊃ ln-1,T ⊃ • • • ⊃ ln-k+1,T of filtrations parametrized by F (k) ln,T . First we consider the case a 2 (λ (n-k+1) ) = 0. We take an affine open covering {U i } i of F (k) ln,T so that we may take two elements ṽi 1 and ṽi 2 of ln-k+1

| U i such that ln-k+1,T | U i = ṽi 1 , ṽi 2 O n[t] with length ṽi 1 O n[t] ≥ length ṽi 2 O n[t] . For each U i , we set li n-k := α i 1,k+1 ṽi 1 + α i 2,k+1 ṽi 2 , f ṽi 1 , f ṽi 2 O n[t]
.

where [α i 1,k+1 : α i 2,k+1 ] ∈ P 1 . We glue {U i × P 1 } i so that these families patch together. Then we have a P 1 -bundle over

F (k) ln,T . If a 2 (λ (n-k+1) ) = a 2 (λ (n-k) ), then we define F (k+1)
ln,T by the 0-section of this P 1 -bundle and define ln-k,T by the gluing of ṽi

2 , f ṽi 1 O n[t] . If a 2 (λ (n-k+1) ) = a 2 (λ (n-k) ) + 1, then we define F (k+1)
ln,T by this P 1 -bundle and define ln-k,T by the gluing of li n-k . Second we consider the case a 2 (λ (n-k+1) ) = 0. In particular a 2 (λ (n-k+1) ) = a 2 (λ (n-k) ). We take an affine open covering {U i } i of F (k) ln,T so that we may take an element ṽi such that ln-k+1,T

| U i = ṽi O n[t] . For each U i , we set li n-k := f ṽi 1 O n[t]
. We define F So the parameter space of refined parabolic structures with fixed

l n is T ∈T ln F (n) ln,T . Remark that F (n) ln,T is irreducible and the dimension of F (n) ln,T is a 2 for each T . If a 2 = 0 or a 2 = 1, we can describe T ∈T ln F (n) ln,T easily. First if a 2 = 0, that is, λ (n) = (1 n , 2 0 ), then #T ln = 1 and F (n) ln,T is a point. So T ∈T ln F (n)
ln,T is a point. This point corresponds to the filtration

l n ⊃ f • l n ⊃ f 2 • l n ⊃ • • • ⊃ f n-1 • l n ⊃ 0.
Next we consider the case a 2 = 1, that is, λ (n) := (0, 1, . . . , 1, 2) = (1 n-2 , 2 1 ). We define a standard tableau T k so that the corresponding sequence of Young diagrams is ln,T k corresponding to T k is the projective line P 1 . The parameter space of refined parabolic structures with fixed l n is

(3.4) λ (n) = (1 n-2 , 2 1 ) ⊃ (1 n-3 , 2 1 ) ⊃ • • • ⊃ (1 n-k-1 , 2 1 ) ⊃ (1 n-k , 2 0 ) ⊃ (1 n-k-1 , 2 0 ) ⊃ • • • ⊃ (1 1 , 2 0 ) for k = 1, 2, . . . ,
F (n) ln,T1 ∪ F (n) ln,T2 ∪ • • • ∪ F (n) ln,Tn-1 ,
which is a chain of (n-1) projective lines. The intersection

F (n) ln,T k 1 ∩ F (n) ln,T k 2 is a point if |k 1 -k 2 | = 1. When |k 1 -k 2 | = 1,
this intersection is empty.

Stability of refined parabolic bundles

In the previous section, we have defined refined parabolic bundles. We would like to consider a moduli space of refined parabolic bundles. In order to obtain a good moduli space, we have to introduce a stability condition of refined parabolic bundles. For this purpose, first, we define the stability index as in [START_REF] Loray | Lagrangian fibrations in duality on moduli spaces of rank 2 logarithmic connections over the projective line[END_REF]Section 2]. This stability condition is the same as the stability condition as in [START_REF] Yokogawa | Compactification of moduli of parabolic sheaves and moduli of parabolic Higgs sheaves[END_REF]. In particular, the notion of parabolic bundles is contained in the notion of refined parabolic bundles. So by the definition of the stability condition for refined parabolic bundles, we have a stability condition for parabolic bundles.

Let (E, {l i,• } i∈I ) be a refined parabolic bundle. We fix weights w = (w 1 , . . . , w ν ),

w i = (w i,ni , . . . , w i,1 ) ∈ [0, 1] ni such that (4.1) 0 ≤ w i,ni ≤ • • • ≤ w i,1 ≤ 1
for any i ∈ I.

Definition 28. Let L be a line subbundle of E. We define the w-stability index of L to the real number

Stab w (L) := deg(E) -2 deg(L) + i∈I ni k=1 w i,k 1 -2 length((l i,k ∩ L| ni[ti] )/(l i,k-1 ∩ L| ni[ti] ))
where l i,0 = 0 for any i.

If we set

Ĩ := {(i, k) | i ∈ I, k = 1, 2, . . . , n i } and ǫ i,k (L) := -1 when length((l i,k ∩ L| ni[ti] )/(l i,k-1 ∩ L| ni[ti] )) = 0 1 when length((l i,k ∩ L| ni[ti] )/(l i,k-1 ∩ L| ni[ti]
)) = 0 , then we have the following equality:

Stab w (L) = deg(E) -2 deg(L) + (i,k)∈ Ĩ ǫ i,k (L) • w i,k .
Definition 29. A refined parabolic bundle (E, l) is w-stable (resp. w-semistable) if for any subbundle L ⊂ E, the following inequality holds:

Stab w (L) > 0 (resp. ≥ 0). Now we discuss on the stability condition by using the parabolic degree. First, we recall a definition of the parabolic degree. We fix parabolic weights α = (α 1 , . . . , α ν ), where each α i is a tuple of real numbers α i = (α i,ni+1 , α i,ni , . . . , α i,1 ) with the condition 0 < α i,ni+1 < α i,ni < • • • < α i,1 < 1. We define an α-parabolic degree of a line subbbundle

L of E as para-deg α (L) = deg(L) + i∈I ni+1 k=1 α i,k length((l i,k ∩ L| ni[ti] )/(l i,k-1 ∩ L| ni[ti] )),
where l i,ni+1 := E| ni [ti] . We can define the stability of refined parabolic bundles by using the parabolic degree: A refined parabolic bundle (E, l) is stable if for any subbundle L ⊂ E, the following inequality holds:

para-deg α (E) rank(E) > para-deg α (L)
rank(L) . Now we will discuss a relation between the stability defined by the parabolic degree and the stability defined by the stability index. We compute para-deg α (E)/rank(E) -para-deg α (L)/rank(L) as follows.

para-deg α (E)

rank(E) - para-deg α (L) rank(L) = 1 2 deg(E) -2 deg(L) + i∈I ni+1 k=1 α i,k length((l i,k )/(l i,k-1 )) -2 i∈I ni+1 k=1 α i,k length((l i,k ∩ L| ni[ti] )/(l i,k-1 ∩ L| ni[ti] )) = 1 2 deg(E) -2 deg(L) + i∈I α i,ni+1 n i + ni k=1 α i,k -2 i∈I α i,ni+1 n i - ni k=1 length((l i,k ∩ L| ni[ti] )/(l i,k-1 ∩ L| ni[ti] )) + ni k=1 α i,k length((l i,k ∩ L| ni[ti] )/(l i,k-1 ∩ L| ni[ti] )) = 1 2 deg(E) -2 deg(L) + i∈I ni k=1 (α i,k -α i,ni+1 ) -2 i∈I ni k=1 (α i,k -α i,ni+1 ) length((l i,k ∩ L| ni[ti] )/(l i,k-1 ∩ L| ni[ti] )) . So if we set w i,k := α i,k -α i,mi+1
, then we have the following equality

para-deg α (E) rank(E) - para-deg α (L) rank(L) = Stab w (L) 2 .
Then the stability defined by the stability index is the same as the stability defined by the parabolic degree.

4.1. Tame and undecomposable refined parabolic bundles. In Section 2, we have defined undecomposable parabolic bundles and admissible parabolic bundles. Here we define undecomposable refined parabolic bundles and admissible refined parabolic bundles.

Definition 30. Let (E, {l i,• } i∈I ) be a refined parabolic bundle of rank 2 and of degree d. We say that (E, {l i,• } i∈I ) is decomposable if there exists a decomposition E = L 1 ⊕ L 2 , where L 1 and L 2 are non trivial, and

l i,k = l (1) i,k ⊕ l (2)
i,k for any i = 1, . . . , ν and k = 1, . . . , n i , where we set l

(1) i,k := l i,k ∩ L 1 | ni[ti] and l (2) i,k := l i,k ∩ L 2 | ni[ti] . We say that (E, {l i,• } i∈I ) is undecomposable if (E, {l i,• } i∈I ) is not decomposable.
Definition 31. Let (E, {l i,• } i∈I ) be a refined parabolic bundle of rank 2 and of degree d. We say (E, {l i,• } i∈I ) is admissible if this refined parabolic bundle satisfies the following condition:

i∈I ni k=1 length((l i,k ∩ L| ni[ti] )/(l i,k-1 ∩ L| ni[ti] )) ≤ n + deg(E) -2 deg(L) -2 for any line subbundles L such that deg(E) ≤ 2 deg(L).
If D is a reduced effective divisor, then we have the fact that (E, {l i,• } i∈I ) is stable for a convenient choice of weights w if, and only if, it is undecomposable (see [START_REF] Loray | Lagrangian fibrations in duality on moduli spaces of rank 2 logarithmic connections over the projective line[END_REF]Proposition 3.4]). On the other hand, when D is not necessarily reduced, there exist examples of (refined) parabolic bundles (E, {l i,• } i∈I ) such that it is undecomposable and it is not stable for any weights w.

Example 32. Let D and (E, l) be one of the four listed parabolic bundles in Lemma 13. Then (E, l) is undecomposable and admissible. But, by the inequalities (4.1), the parabolic bundle (E, l) is not stable for any weights w for any cases in Lemma 13. In particular, there exist parabolic bundles (E, {l i } i∈I ) such that it is Λ-flat and that it is not stable for any weights w. (Here Λ is defined as in Proposition 14). Now we will find a necessary and sufficient condition of a refined parabolic bundle (E, {l i,• } i∈I ) for the condition that there exist weights w such that (E, {l i,• } i∈I ) is w-stable. For this purpose, we will define tame refined parabolic bundles. For a line subbundle L ⊂ E and i ∈ I, we define an integer N i (L) as

(4.2) N i (L) = max k ′ ∈{1,2,...,ni}    k ′ k=1 ǫ i,k (L)    .
We set

I + L = {i ∈ I | N i (L) > 0}. Moreover, we set k i,max (L) := max    k ′ ∈ {1, 2, . . . , n i } N i (L) = k ′ k=1 ǫ i,k (L)    , K i,max (L) := {1, 2, . . . , k i,max (L)}, and K c i,max (L) := {1, 2, . . . , n i } \ K i,max (
L). Definition 33. Let (E, {l i,• } i∈I ) be a refined parabolic bundle of rank 2 and of degree d. We say (E, {l i,• } i∈I ) is tame if this refined parabolic bundle satisfies the following condition: When d 1 = d 2 , there also exists a line subbundle L such that deg(E) ≤ 2 deg(L). Remark that this line bundle is not unique in this case. Now, we will show the implication (1.2), that is,

• I + L is
(E, {l i,• } i∈I ) is undecomposable and tame =⇒ (E, {l i,• } i∈I ) is admissible.
Moreover we will show that, when (E, l) is a parabolic bundle, (E,l) is simple ⇐⇒ (E, l) is undecomposable and tame.

Proposition 34. If a refined parabolic bundle (E, {l i,• } i∈I ) is tame and undecomposable, then it is admissible.

Proof. For each i ∈ I + L , we have an inequality

N i (L) ≤ k∈{k|ǫ i,k (L)=1} ǫ i,k (L) = n i - ni k=1 length((l i,k ∩ L| ni[ti] )/(l i,k-1 ∩ L| ni[ti] )).
Clearly

n i - ni k=1 length((l i,k ∩ L| ni[ti] )/(l i,k-1 ∩ L| ni[ti]
)) ≥ 0. Then we have that for any line subbundles L such that deg(E) ≤ 2 deg(L),

-deg(E) + 2 deg(L) + 1 ≤ i∈I + L N i (L) ≤ i∈I + L n i - ni k=1 length((l i,k ∩ L| ni[ti] )/(l i,k-1 ∩ L| ni[ti] )) ≤ i∈I n i - ni k=1 length((l i,k ∩ L| ni[ti] )/(l i,k-1 ∩ L| ni[ti] )) We set E = O(d 1 ) ⊕ O(d 2 ) (with d 1 ≤ d 2 ) and L = O(d 2 ). We assume that -deg(E) + 2 deg(L) + 1 = i∈I + L N i (L), (4.3) N i (L) = k∈{k|ǫ i,k (L)=1} ǫ i,k (L) (for any i ∈ I + L ), (4.4) and (4.5) n i - ni k=1 length((l i,k ∩ L| ni[ti] )/(l i,k-1 ∩ L| ni[ti] )) = 0 (for any i ∈ I + L ).
By the equalities (4.4) and (4.5), we have

(4.6) ǫ i,k (L) =      1
for any i ∈ I + L and any k ∈ K i,max (L) -1 for any i ∈ I + L and any k ∈ K c i,max (L) -1 for any i ∈ I + L and any k ∈ {1, 2, . . . , n i }

.

By the equality (4.3), we have the following equality:

(4.7) d 2 -d 1 + 1 = i∈I + L #K i,max (L) = i∈I + L k i,max (L). If d 1 = d 2 , then i∈I + L k i,max (L) = 1. So #I + L = 1 and K i0,max (L) = {1} for the i 0 ∈ I + L . We have that ǫ i,k (L) = 1 when (i, k) = (i 0 , 1) (with i 0 ∈ I + L ) and ǫ i,k (L) = -1 otherwise. This implies that (E, {l i,• } i∈I ) is decomposable. It is a contradiction. So d 1 = d 2 .
(d 1 ) ⊕ O(d 2 ) such that l i,k = l (1) i,k ⊕ l (2)
i,k for any i = 1, . . . , ν and k = 1, . . . , n i . Here we set l

(1) i,k := l i,k ∩ O(d 1 )| ni[ti] and l (2) i,k := l i,k ∩ O(d 2 )| ni[ti] . Since (E, {l i,• } i∈I ) is undecomposable, it is a contradiction. So we have -deg(E) + 2 deg(L) + 1 < i∈I n i - ni k=1 length((l i,k ∩ L| ni[ti] )/(l i,k-1 ∩ L| ni[ti] )) .
This inequality implies that (E, {l i,• } i∈I ) is admissible.

Proposition 35. Let (E, {l i } i∈I ) be a parabolic bundle (Definition 7) and (E, {l i,• } i∈I ) be the corresponding refined parabolic bundle. The parabolic bundle (E, {l i } i∈I ) is simple if, and only if, (E, {l i,• } i∈I ) is undecomposable and tame.

Proof. First we will show that (E, {l i,• } i∈I ) is undecomposable and tame when (E, {l i } i∈I ) is simple. If (E, {l i } i∈I ) is simple, (E, {l i,• } i∈I ) is undecomposable by Proposition 9. So we will show that (E, {l i } i∈I ) is not simple when (E, {l i,• } i∈I ) is not tame. We will show that there is a non-zero nilpotent element of End(E, {l i } i∈I ). Since (E, {l i,• } i∈I ) is not tame, there exists a line subbundle

L 0 with deg(E) ≤ deg(L 0 ) such that I + L0 is empty or -deg(E) + 2 deg(L 0 ) ≥ i∈I + L 0 N i (L 0 ). We decompose E = L ⊕ L 0 with deg(L) ≤ deg(L 0 )
. We take a nilpotent element N ∈ End(L ⊕ L 0 ) as follows:

N = 0 0 f 0 where f = f (x) is a polynomial where deg(f ) ≤ 2 deg(L 0 ) -deg(E). We set m i = length(l i ∩ O(L 0 )| ni[ti]
) and

I + = {i ∈ I | n i -2m i ≥ 0}.
Since l i is free, we may check that

ǫ i,k (L 0 ) = 1 (m i < k ≤ n i ) -1 (1 ≤ k ≤ m i )
for any i ∈ I. So we have that N i (L 0 ) = n i -2m i and I + = I + L0 . First we assume that I + L0 is empty. Then, by using the equalities (2.2), we may check that there exists a non-zero polynomial f such that N preserves the parabolic structure {l i } i∈I . So if I + L0 is empty, then (E, {l i } i∈I ) is not simple. Second we assume that

I + = ∅ and -deg(E) + 2 deg(L 0 ) ≥ i∈I + L 0 N i (L 0 )
. By this inequality, we have that (4.8)

i∈I + (n i -2m i ) ≤ 2 deg(L 0 ) -deg(E).
Then, by using the equalities (2.2), we may check that there exists a non-zero polynomial f such that N preserves the parabolic structure {l i } i∈I . Finally we obtain that (E, {l i } i∈I ) is not simple when (E, {l i,• } i∈I ) is not tame.

Next we will show that (E, {l i } i∈I ) is simple when (E, {l i,• } i∈I ) is undecomposable and tame. Since (E, {l i,• } i∈I ) is undecomposable, any endomorphism A of the (E, {l i } i∈I ) takes the form A = c • Id E + N with c ∈ C and N nilpotent. We assume that there exists a non-zero nilpotent element N of End(E, {l i } i∈I ). By using the argument as in the verification of the inequality (2.3), we may check that there exists a line subbundle

L 0 with deg(E) ≤ 2 deg(L 0 ) such that i∈I + (n i -2m i ) ≤ 2 deg(L 0 ) -deg(E).
Here, m i and I + are defined as above for this L 0 . By the argument as above, we have that N i (L 0 ) = n i -2m i and I + L0 = I + . So we obtain that

I + L0 is empty or the inequality -deg(E) + 2 deg(L 0 ) ≥ i∈I + L 0 N i (L 0 ) holds. Since (E, {l i,• } i∈I ) is tame, it is a contradiction. So (E, {l i } i∈I ) is simple.

Proof of Theorem A. Now, we start to show Theorem A (Theorem 1 and Corollary 2 below).

Proposition 36. If a refined parabolic bundle (E, {l i,• } i∈I ) is stable for a convenient choice of weights w, then it is tame and undecomposable.

Proof. Assume that (E, {l i,• } i∈I ) is decomposable. There exists a decomposition E ∼ = L 1 ⊕ L 2 such that Stab w (L 1 ) + Stab w (L 2 ) = 0 for any weights w. Then (E, {l i,• } i∈I ) is not stable for any weights w. Assume that (E, {l i,• } i∈I ) is not tame. There exists a line subbundle L ⊂ E with deg(E) ≤ 2 deg(L) such that

• I + L is empty, or • I + L is not empty and -deg(E) + 2 deg(L) ≥ i∈I + L N i (L).
If I + L is empty, then we have the inequality ni k=1 ǫ i,k (L) • w i,k ≤ 0 for any i ∈ I, since we have the inequalities (4.1). So we have

Stab w (L) = deg(E) -2 deg(L) + i∈I ni k=1 ǫ i,k (L) • w i,k ≤ deg(E) -2 deg(L) ≤ 0.
Then (E, {l i,• } i∈I ) is not stable for any weights w. Now we assume that I + L is not empty. By the inequalities (4.1) and the definitions of K i,max (L) and K c i,max (L), we have the following inequalities for each i ∈ I + L :

k∈K c i,max (L) ǫ i,k (L) • w i,k ≤ 0 and k∈Ki,max(L) ǫ i,k (L) • w i,k ≤ N i (L). If i ∈ I + L , then ni k=1 ǫ i,k (L) • w i,k ≤ 0.
By these inequalities, we have

Stab w (L) = deg(E) -2 deg(L) + i∈I\I + L ni k=1 ǫ i,k (L) • w i,k + i∈I + L k∈K c i,max (L) ǫ i,k (L) • w i,k + i∈I + L k∈Ki,max(L) ǫ i,k (L) • w i,k ≤ deg(E) -2 deg(L) + i∈I + L N i (L) ≤ 0.
Then (E, {l i,• } i∈I ) is not stable for any weights w.

Proposition 37. Assume that

E = O(d 1 ) ⊕ O(d 2 ) with d 1 < d 2 .
If a refined parabolic bundle (E, {l i,• } i∈I ) is tame and undecomposable, then it is stable for a convenient choice of weights w.

Proof. We assume that (E, {l i,• } i∈I ) is tame and undecomposable. Since d 1 < d 2 , there exists a unique morphism ϕ d2 : O(d 2 ) ֒→ E. We set

N c i (O(d 2 )) := k∈K c i,max (O(d2)) ǫ i,k (O(d 2 )) for i ∈ I + O(d2) ni k=1 ǫ i,k (O(d 2 )) for i ∈ I + O(d2)
.

For the unique morphism ϕ d2 : O(d 2 ) ֒→ E, we define weights w = (w 1 , . . . , w n ) as follows:

w i,k =      w for i ∈ I + O(d2) and k ∈ K i,max (O(d 2 )) w ′ for i ∈ I + O(d2) and k ∈ K c i,max (O(d 2 )) w ′ for i ∈ I + O(d2) and k ∈ {1, 2, . . . , n i } .
Here w and w ′ are real numbers such that (4.9)

     w > - i∈I N c i (O(d2)) i∈I + O(d 2 ) Ni(O(d2)) • w ′ + d2-d1 i∈I + O(d 2 ) Ni(O(d2) w < -i∈I N c i (O(d2))-2 i∈I + O(d 2 ) Ni(O(d2)) • w ′ + d2-d1 i∈I + O(d 2 )
Ni(O(d2)) and 0 < w ′ < w < 1 w + w ′ < 1 .

Remark that, since (E, {l i,• } i∈I ) is tame and d 1 < d 2 , we have the following inequalities:

0 < d 2 -d 1 i∈I + O(d 2 ) N i (O(d 2 )) < 1 and - i∈I N c i (O(d 2 )) i∈I + O(d 2 ) N i (O(d 2 )) < -i∈I N c i (O(d 2 )) -2 i∈I + O(d 2 ) N i (O(d 2 )
) .

So there exist such real numbers w and w ′ . We will check that (E, {l i,• } i∈I ) is w-stable for such a weights w. For the subbundle O(d 2 ) ⊂ E, we have the following inequality:

Stab w (O(d 2 )) = d 1 -d 2 + i∈I N c i (O(d 2 )) w ′ +    i∈I + O(d 2 ) N i (O(d 2 ))    w > 0
by the inequalities (4.9). Next we calculate the stability index Stab w (O(d 1 )) for any subbundles O(d 1 ) ⊂ E:

Stab w (O(d 1 )) ≥ d 2 -d 1 - i∈I N c i (O(d 2 )) w ′ -    i∈I + O(d 2 ) N i (O(d 2 ))    w + 2w ′ > 0
by the inequalities (4.9). Here in the first inequality, we used the assumption that (E, {l i,• } i∈I ) is undecomposable and w ′ < w. Let L be a subbundle L ⊂ E such that deg(L) ≤ d 1 -1. We will calculate the stability index Stab w (L). For the subbundle L, we set

K (i,d2) δ1,δ2 (L) := {k | k ∈ {1, 2, . . . , n i }, and ǫ i,k (L) = δ 1 , ǫ i,k (O(d 2 )) = δ 2 },
where i ∈ I and δ 1 , δ 2 ∈ {+, -}. We set k

(i,d2) δ1,δ2 (L) = #K (i,d2)
δ1,δ2 (L) for any i ∈ I, and

k (i,d2) δ1,δ2 (L) ′ = #(K (i,d2) δ1,δ2 (L) ∩ K i,max (O(d 2 ))) and k (i,d2) δ1,δ2 (L) ′′ = #(K (i,d2) δ1,δ2 (L) ∩ K c i,max (O(d 2 )))
for any i ∈ I + L . We have the following inequality:

i∈I + O(d 2 ) ni k=1 ǫ i,k (L) • w i,k + i∈I\I + O(d 2 ) ni k=1 ǫ i,k (L) • w ′ = i∈I + O(d 2 )
(k

(i,d2) +,+ (L) ′ -k (i,d2) -,+ (L) ′ + k (i,d2) +,-(L) ′ -k (i,d2) -,-(L) ′ ) • w + i∈I + O(d 2 )
(k

(i,d2) +,+ (L) ′′ -k (i,d2) -,+ (L) ′′ + k (i,d2) +,-(L) ′′ -k (i,d2) -,-(L) ′′ ) • w ′ + i∈I\I + O(d 2 )
(k

(i,d2) +,+ (L) -k (i,d2) -,+ (L) + k (i,d2) +,-(L) -k (i,d2) -,-(L)) • w ′ ≥ i∈I + O(d 2 ) (-N i (O(d 2 )) -2k (i,d2) -,-(L) ′ ) • w + i∈I + O(d 2 ) (-N c i (O(d 2 )) -2k (i,d2) -,-(L) ′′ ) • w ′ + i∈I\I + O(d 2 ) (-N c i (O(d 2 )) -2k (i,d2) -,-(L)) • w ′
By this inequality and w ′ < w, we have the following inequality:

(4.10)

Stab w (L) ≥ deg(E) -2 deg(L) - i∈I N c i (O(d 2 ))w ′ - i∈I + O(d 2 ) N i (O(d 2 ))w -2 i∈I k (i,d2) -,-(L) w
We describe the subbundle

L ֒→ O(d 1 ) ⊕ O(d 2 ) by a vector (f L 1 (x), f L 2 (x)), where f L 1 (x) and f L 2 (x) are polynomials in x such that deg(f L 1 (x)) ≤ d 1 -deg(L) and deg(f L 2 (x)) ≤ d 2 -deg(L). We have the zero order of f L 1 (x) at t i ≥ #{k ∈ {1, 2, . . . , n i } | ǫ i,k (L) = ǫ i,k (O(d 2 )) = -1}
for each i ∈ I. By this inequality, we have

d 1 -deg(L) ≥ i∈I k (i,d2) -,-(L). When i∈I k (i,d2)
-,-(L) = 0 for the subbundle L ⊂ E, we have the following inequality:

Stab w (L) ≥ d 2 -d 1 - i∈I N c i (O(d 2 ))w ′ - i∈I + O(d 2 ) N i (O(d 2 ))w + i∈I k (i,d2) -,-(L) • 2(1 -w) ≥ d 2 -d 1 - i∈I N c i (O(d 2 )) w ′ -    i∈I + O(d 2 ) N i (O(d 2 ))    w + 2(1 -w) > 2(1 -w -w ′ ) > 0.
Here in the first inequality, we used the inequality

d 1 -deg(L) ≥ i∈I k (i,d2)
-,-(L) and the inequality (4.10). In the second inequality, we used the inequalities i∈I k (i,d2) -,-(L) ≥ 1 and w < 1. In the third inequality and in the last inequality, we used the inequalities in (4.9). When i∈I k

(i,d2) -,-(L) = 0,
for the subbundle L ⊂ E, we have the following inequality:

Stab w (L) ≥ d 1 + d 2 -2 deg(L) - i∈I N c i (O(d 2 )) w ′ -    i∈I + O(d 2 ) N i (O(d 2 ))    w ≥ d 2 -d 1 - i∈I N c i (O(d 2 )) w ′ -    i∈I + O(d 2 ) N i (O(d 2 ))    w + 1 > 1 -2w ′ > 0.
Here we used the inequality deg(L) ≤ d 1 -1 in the second inequality. Then we obtain that (E, {l i,• } i∈I ) is w-stable for the weights w.

Proposition 38. Assume that E = O(d) ⊕ O(d).
If a refined parabolic bundle (E, {l i,• } i∈I ) is tame and undecomposable, then it is stable for a convenient choice of weights w.

Proof. We take an element i 1 ∈ I. Since E = O(d) ⊕ O(d) and length(l i1,1 ) = 1, we can take a line subbundle

L 1 ⊂ E with deg(L 1 ) = d such that ǫ i1,1 (L 1 ) = -1. Since (E, {l i,• } i∈I ) is tame, there exists an element i 2 ∈ I such that N i2 (L 1 ) = k∈Ki 2 ,max (L1) ǫ i2,k ≥ 1.
We divide into the following two cases:

(A) the set {k ∈ K i2,max (L 1 ) | ǫ i2,k (L 1 ) = 1, ǫ i2,k (L) = -1} is empty for any line subbundles L ⊂ E with deg(L) = d, (B) the set {k ∈ K i2,max (L 1 ) | ǫ i2,k (L 1 ) = 1, ǫ i2,k (L) = -1} is not empty for some line subbundle L ⊂ E with deg(L) = d.
For the case (A), we determine weights w as follows:

(

A-i) When i 1 = i 2 , w i,k :=      w 1 when i = i 1 and k = 1 w 2 when i = i 2 and k ∈ K i2,max (L 1 ) 0 otherwise . (A-ii) When i 1 = i 2 , w i,k :=      w 1 when i = i 1 and k = 1 w 2 when i = i 1 and k ∈ K i1,max (L 1 ) \ {1} 0 otherwise with 0 < w 2 < w 1 < 1.
For the case (B), we determine weights w as follows. We take a line subbundle

L 2 of E with deg(L 2 ) = d so that (4.11) #{k ∈ K i2,max (L 1 ) | ǫ i2,k (L 1 ) = 1, ǫ i2,k (L 2 ) = -1} is maximized. In particular L 1 = L 2 in E = O(d) ⊕ O(d). So it is impossible that ǫ i,k (L 1 ) = ǫ i,k (L 2 ) = -1 for any (i, k) ∈ Ĩ. Since (E, {l i,• } i∈I ) is undecomposable, the set (4.12) {(i, k) ∈ Ĩ | ǫ i,k (L 1 ) = ǫ i,k (L 2 ) = 1}
is not empty. We take an element (i 3 , k 3 ) of the set (4.12) such that the pair satisfies the equality

k 3 = min{k ∈ {1, 2, . . . , n i3 } | ǫ i3,k (L 1 ) = ǫ i3,k (L 2 ) = 1}. If the subset (4.13) {k ∈ {1, 2, . . . , n i } | ǫ i2,k (L 1 ) = ǫ i2,k (L 2 ) = 1} ∩ K i2,max (L 1 )
is not empty, we take the element (i 3 , k 3 ) from this subset (4.13). So, if i 2 = i 3 , we have that

(4.14) ǫ i2,k (L 1 ) + ǫ i2,k (L 2 ) = 0 (k ∈ K i2,max (L 1 )).
So, when i 2 = i 3 , or when i 2 = i 3 and k 3 > k i2,max (L 1 ), we have that

(4.15) ǫ i2,k (L) = 1 (k ∈ K i2,max (L 1 ))
for any line subbundles L such that deg(L) = d, L = L 1 , and L = L 2 .

(B-i) When i 1 , i 2 , i 3 are distinct from each other, we determine weights w as follows:

w i,k :=                w 1 when i = i 1 and k = 1 w 2 when i = i 2 and k ∈ K i2,max (L 1 ) w 3 when i = i 3 and k = 1, 2, . . . , k 3 -1 w 4 when i = i 3 and k = k 3 0 otherwise with 0 < w 1 , w 2 < 1 and 0 < w 4 < w 3 < 1. (B-ii) When i 1 = i 2 = i 3 and k i2,max (L 1 ) ≥ k 3 ,
we determine weights w as follows:

w i,k :=      w 1 when i = i 1 and k = 1 w 2 when i = i 2 and k ∈ K i2,max (L 1 ) 0 otherwise with 0 < w 1 , w 2 < 1. (B-iii) When i 1 = i 2 = i 3 and k 3 > k i1,max (L 1 )
, we determine weights w as follows:

w i,k :=                w 1 when i = i 1 and k = 1 w 2 when i = i 2 and k ∈ K i2,max (L 1 ) w 3 when i = i 2 and k = k i2,max (L 1 ) + 1, . . . , k 3 -1 w 4 when i = i 2 and k = k 3 0 otherwise with 0 < w 4 < w 3 < w 2 < 1 and 0 < w 1 < 1. (B-iv) When i 1 = i 3 = i 2 ,
we determine weights w as follows:

w i,k :=                w 1 when i = i 1 and k = 1 w 2 when i = i 2 and k ∈ K i2,max (L 1 ) w 3 when i = i 1 and k = 2, 3, . . . , k 3 -1 w 4 when i = i 1 and k = k 3 0 otherwise with 0 < w 2 < 1 and 0 < w 4 < w 3 < w 1 < 1.
(B-v) When i 1 = i 2 = i 3 , we determine weights w as follows:

w i,k :=                w 1 when i = i 1 and k = 1 w 2 when i = i 1 and k ∈ K i1,max (L 1 ) \ {1} w 3 when i = i 3 and k = 1, . . . , k 3 -1 w 4 when i = i 3 and k = k 3 0 otherwise with 0 < w 2 < w 1 < 1 and 0 < w 4 < w 3 < 1. (B-vi) When i 1 = i 2 = i 3 and k i1,max (L 1 ) ≥ k 3 ,
we determine weights w as follows:

w i,k :=      w 1 when i = i 1 and k = 1 w 2 when i = i 1 and k ∈ K i1,max (L 1 ) \ {1} 0 otherwise with 0 < w 2 < w 1 < 1. (B-vii) When i 1 = i 2 = i 3 and k 3 > k i1,max (L 1 )
, we determine weights w as follows:

w i,k :=                w 1 when i = i 1 and k = 1 w 2 when i = i 1 and k ∈ K i1,max (L 1 ) \ {1} w 3 when i = i 1 and k = k i1,max (L 1 ) + 1, . . . , k 3 -1 w 4 when i = i 1 and k = k 3 0 otherwise with 0 < w 4 < w 3 < w 2 < w 1 < 1.
If we take a convenient choice of w 1 , w 2 , w 3 , w 4 (or w 1 , w 2 ), we may check that (E, {l i,• } i∈I ) is w-stable for the weights w. Now we check this claim for the 3 cases: (A-i); (B-i); and (B-vi). For other cases, we will check (E, {l i,• } i∈I ) is w-stable for some weights w by the same argument as in the three cases.

We consider the case (A-i). We assume that

-w 1 + N i2 (L 1 ) • w 2 > 0.
Since N i2 (L 1 ) ≥ 1, there exists a pair w 1 , w 2 satisfies this inequality and 0 < w 2 , w 1 < 1. We have the following inequality for Stab w (L 1 ):

Stab w (L 1 ) = -w 1 + N i2 (L 1 ) • w 2 > 0.
For any line subbundles L ⊂ E with L 1 = L, we have the following inequality:

Stab w (L) = deg(E) -2 deg(L) + w 1 + k i2,max (L 1 ) • w 2 ≥ w 1 + k i2,max (L 1 ) • w 2 > 0.
Then we obtain that (E, {l i,• } i∈I ) is w-stable for the weights w.

We consider the case (B-i). We assume that

           -w 1 + N i2 (L 1 )w 2 + k3-1 k ′ =1 ǫ i3,k ′ (L 1 ) w 3 + w 4 > 0 -w 1 + N i2 (L 1 )w 2 + k3-1 k ′ =1 ǫ i3,k ′ (L 1 ) w 3 -w 4 < 0 w 1 + k i2,max (L 1 ) • w 2 + (k 3 -1)w 3 -w 4 > 0 w 1 + k i2,max (L 1 ) • w 2 + (k 3 -1)w 3 + w 4 < 2 .
We may check that there exist such real numbers w 1 , w 2 , w 3 , w 4 . We have the following inequalities for Stab w (L 1 ) and Stab w (L 2 ):

Stab w (L 1 ) = -w 1 + N i2 (L 1 )w 2 + k3-1 k ′ =1 ǫ i3,k ′ (L 1 ) w 3 + w 4 > 0 Stab w (L 2 ) = w 1 -N i2 (L 1 )w 2 - k3-1 k ′ =1 ǫ i3,k ′ (L 1 ) w 3 + w 4 > 0.
Here for the equality in the computation of Stab w (L 2 ), we used the equality (4.14) and the equality

ǫ i3,k (L 1 ) + ǫ i3,k (L 2 ) = 0 for k = 1, 2, . . . , k 3 -1.
This equality is given by the equality

k 3 = min{k ∈ {1, 2, . . . , n i3 } | ǫ i3,k (L 1 ) = ǫ i3,k (L 2 ) = 1}. For any line subbundles L such that deg(L) = d, L = L 1 ,
and L = L 2 , we have the following inequalities:

Stab w (L) ≥ w 1 + k i2,max (L 1 ) • w 2 + (k 3 -1)w 3 -w 4 > 0.
by the equality (4.15). For any line subbundles L with deg(L) ≤ d -1, we have the following inequalities

Stab w (L) ≥ deg(E) -2 deg(L) -w 1 -k i2,max (L 1 ) • w 2 -(k 3 -1)w 3 -w 4 ≥ 2 -w 1 -k i2,max (L 1 ) • w 2 -(k 3 -1)w 3 -w 4 > 0.
Then we obtain that (E, {l i,• } i∈I ) is w-stable for the weights w.

We consider the case (B-vi). We assume that

     -w 1 + (N i2 (L 1 ) + 1) • w 2 > 0 w 1 -(N i2 (L 1 ) -1) • w 2 > 0 w 1 + (k i2,max (L 1 ) -1) • w 2 < 2 .
Since N i2 (L 1 ) ≥ 1, there exists a pair w 1 , w 2 satisfies this inequality and 0 < w 2 , w 1 < 1. We have the following inequality for Stab w (L 1 ):

Stab w (L 1 ) = -w 1 + (N i2 (L 1 ) + 1) • w 2 > 0, since ǫ i1,1 (L 1 ) = -1 and N i2 (L 1 ) = k∈Ki 2 ,max (
Stab w (L) ≥ Stab w (L 2 ) ≥ deg(E) -2 deg(L 2 ) + w 1 + (-N i2 (L 1 ) + 1) • w 2 = w 1 -(N i2 (L 1 ) -1) • w 2 > 0.
Here for the first inequality, we used the fact that L 2 maximizes #{k ∈ K i2,max (L 1 ) | ǫ i2,k (L 1 ) = 1, ǫ i2,k (L 2 ) = -1} and for the second inequality we used k i1,max (L 1 ) ≥ k 3 . For any line subbundles L ⊂ E with deg(L) ≤ d -1, we have the following inequality:

Stab w (L) ≥ deg(E) -2 deg(L) -w 1 -(k i2,max (L 1 ) -1) • w 2 ≥ 2 -w 1 -(k i2,max (L 1 ) -1) • w 2 > 0.
Then we obtain that (E, {l i,• } i∈I ) is w-stable for the weights w.

By Proposition 36, Proposition 37, and Proposition 38, we obtain the following theorem: Theorem 1. A refined parabolic bundle (E, {l i,• } i∈I ) is tame and undecomposable if, and only if, it is stable for a convenient choice of weights w.

By Proposition 35, we have the following corollary:

Corollary 2. Let (E, {l i } i∈I ) be a parabolic bundle (Definition 7) and (E, {l i,• } i∈I ) be the corresponding refined parabolic bundle. The parabolic bundle (E, {l i } i∈I ) is simple if, and only if, (E, {l i,• } i∈I ) is stable for a convenient choice of weights w

Elementary transformation of refined parabolic bundles

For studying parabolic connections or parabolic bundles (where D is a reduced effective divisor), the elementary transformations play an important role. In this section, we will define elementary transformations for refined parabolic bundles and will show some properties of the elementary transformations.

5.1. Elementary transformations for Λ-connections. First we recall the elementary transformations for Λ-connections. Let (E, ∇) be a Λ-connection. For (E, ∇) we may define transformations as follows. Let l = {l i } i∈I be the corresponding parabolic structure of (E, ∇). That is, l i is a free submodule of E| ni [ti] . We take an integer k where

1 ≤ k ≤ n i . Let ϕ k : E| ni[ti] → E| k[ti]
be the natural morphism. For the free submodule ϕ k (l i ) of E| k[ti] , the vector bundle E ′ ϕ k (li) is defined by the exact sequence of sheaves Definition 5 for the definition of the formal data). By this transformation, we have a map Con Λ (D) → Con Λ ′ i,k (D).

0 -→ E ′ ϕ k (li) -→ E -→ E| k[ti] /ϕ k (l i ) -→ 0. Note that E| k[ti] /ϕ k (l i ) is a skyscraper sheaf supported on t i . The degree of E ′ ϕ k (li) is deg(E) -k. Let ∇ ϕ k (li) be the induced connection by the morphism E ′ ϕ k (li) → E. So we have a transformation (E ′ ϕ k (li) , ∇ ϕ k (li) ). Let Λ ′ i,k be the formal data of (E ′ ϕ k (li) , ∇ ϕ k (li) ) (see

5.2.

Elementary transformations for parabolic bundles. Second we recall the elementary transformations for parabolic bundles. Let (E, {l i } i∈I ) be a parabolic bundle. For (E, {l i } i∈I ) we may define transformations as follows. For the free submodule l i of E| ni[ti] , the vector bundle E ′ i is defined by the exact sequence of sheaves

0 -→ E ′ i -→ E -→ E| ni[ti] /l i -→ 0. The degree of E ′ i is deg(E) -n i .
We have a filtration of sheaves

E ′ i ⊗ O(-n i [t i ]) ⊂ / / E ⊗ O(-n i [t i ]) ⊂ / / E ′ i ⊂ / / E.
We define a parabolic structure

l ′ i on E ′ i by l ′ i = (E ⊗ O(-n i [t i ]))/(E ′ i ⊗ O(-n i [t i ])). So we have a transformation (E ′ i , {l j } j∈I\{i} ∪ {l ′ i }).
We call this transformation the elementary transformation of a parabolic bundles (E, {l i } i∈I ) at n i [t i ]. We have the elementary transformations for Λ-connections and parabolic bundles:

(E, ∇) → (E ′ ϕ k (li) , ∇ ϕ k (li) ) and (E, {l i } i∈I ) → (E ′ i , {l j } j∈I\{i} ∪ {l ′ i })
, respectively. On the other hand, we have a correspondence (E, ∇) → (E, {l i } i∈I ). If k = n i , then these transformations are compatible with this correspondence. 5.3. Definition of elementary transformations for refined parabolic bundles. Now we consider the extension of the elementary transformation of parabolic bundles. That is, we will define the elementary transformation for refined parabolic bundles. For this purpose, we rephrase refined parabolic structures as filtrations of sheaves. By this rephrasing, we may define the elementary transformation for refined parabolic bundles simply. Let (E, l) be a refined parabolic bundle, where l = {l i,• } i∈I . We take i 0 ∈ I and consider the filtration l i0,• :

E| ni 0 [ti 0 ] ⊃ l i0,ni 0 ⊃ l i0,ni 0 -1 ⊃ • • • ⊃ l i0,1 ⊃ 0. We set E (k) i0 := ker(E -→ E| ni 0 [ti 0 ] /l i0,k )
for k = 0, 1, 2, . . . , n i0 . Then we have a filtration of sheaves (5.1)

E ⊃ E (ni 0 ) i0 ⊃ E (ni 0 -1) i0 ⊃ • • • ⊃ E (1) i0 ⊃ E (0) i0 = E(-n i0 [t i0 ])
where length(E/E

(ni 0 ) i0 ) = n i0 and length(E (k) i0 /E (k-1) i0 ) = 1 for k = 1, 2, . . . , n i0 . Conversely, if we set l i0,k := E (k) i0 /E (0)
i0 for k = 0, 1, 2, . . . , n i0 , then we have a filtration

E/E (0) i0 = E| ni[ti] ⊃ l i0,ni 0 ⊃ l i0,ni 0 -1 ⊃ • • • ⊃ l i0,1 ⊃ 0 of O ni 0 [ti 0 ]
-modules where the length of l i0,k is k. Now we will transform the filtration (5.1) into a new filtration of sheaves. We set

E ′ i0 = E (ni 0 ) i0 . Since length(E (k) i0 /E (k-1) i0 ) = 1 for k = 1, 2, . . . , n i0 , we may check that E (k-1) i0 ⊃ E (k) i0 (-[t i0 ]
) for k = 1, 2, . . . , n i0 . So we can define a filtration of sheaves as

E ′ i0 = E (ni 0 ) i0 ⊃ E (0) i0 ⊃ E (1) i0 (-[t i0 ]) ⊃ E (2) i0 (-2[t i0 ]) ⊃ • • • • • • ⊃ E (ni 0 -1) i0 (-(n i0 -1)[t i0 ]) ⊃ E (ni 0 ) i0 (-n i0 [t i0 ]).
This is a new filtration of sheaves. For k = 0, 1, . . . , n i0 , we set

(E ′ i0 ) (ni 0 -k) := E (k) (-k[t i0 ]
). We will check that length(E ′ i0 /(E ′ i0 ) (ni 0 ) ) = n i0 and length((E ′ i0 ) (k) /(E ′ i0 ) (k-1) ) = 1 for k = 1, 2, . . . , n i0 as follows. For k = 1, 2, . . . , n i0 , we have the following equalities:

length(E (k) i0 /E (k-1) i0 ) + length((E ′ i0 ) (ni 0 -k+1) /(E ′ i0 ) (ni 0 -k) ) = length(E (k) i0 /E (k-1) i0 ) + length(E (k-1) i0 (-(k -1)[t i0 ])/E (k) i0 (-k[t i0 ])) = length(E (k) i0 /E (k) i0 (-[t i0 ])) = 2. Since length(E (k) i0 /E (k-1) i0 ) = 1 for k = 1, 2, . . . , n i0 , we have length((E ′ i0 ) (ni 0 -k+1) /(E ′ i0 ) (ni 0 -k) ) = 1 for k = 1, 2, . . . , n i0 . Moreover we may check that length(E ′ i0 /(E ′ i0 ) (ni 0 ) ) = n i0 . Finally, if we set l ′ i0,k := (E ′ i0 ) (k) /(E ′ i0 ) (0)
for k = 0, 1, 2, . . . , n i0 , then we have a filtration (5.2)

E| ni[ti] ⊃ l ′ i0,ni 0 ⊃ l ′ i0,ni 0 -1 ⊃ • • • ⊃ l ′ i0,1 ⊃ 0 of O ni 0 [ti 0 ] -modules where the length of l ′ i0,k is k. Proposition 43. We take i 0 ∈ I. Let (E ′ i0 , l ′ i0 ) = elm - ni 0 [ti 0 ] (E, {l i,• } i∈I ).
Let L be a line subbundle of E and L ′ be a induced line subbundle of E ′ i0 via E ′ i0 ⊂ E. We set w ′ i,k = 1w i,ni 0 -k+1 for i = i 0 and k = 1, . . . , n i0 w i,k for i = i 0 and k = 1, . . . , n i .

Let Stab w ′ (L ′ ) be the stability index of L ′ with respect to (E ′ i0 , l ′ i0 ) and the weights

w ′ = {w ′ i,k }. Then Stab w (L) = Stab w ′ (L ′ ).
Proof. Since we have a short exact sequence

0 -→ E (ni 0 ) -→ E -→ E/E (ni 0 ) -→ 0 and length(E/E (ni 0 ) ) = n i0 , we have deg(E) = deg(E ′ i0 ) + n i0 .
We have a short exact sequence

0 -→ L ′ -→ L -→ E| L /E (ni 0 ) | L -→ 0. Since length(E| L /E (ni 0 ) | L ) = length(E| L /E(-n i0 [t i0 ])| L ) - ni 0 k=1 length(E (k) | L /E (k-1) | L ) = n i0 - ni 0 k=1 length(E (k) | L /E (k-1) | L ),
we have an equality:

deg(L) = deg(L ′ ) + n i0 - ni 0 k=1 length(E (k) | L /E (k-1) | L ).
Moreover, we have the following equalities

length(E (k) | L /E (k-1) | L ) + length((E ′ i0 ) (ni 0 -k+1) | L ′ /(E ′ i0 ) (ni 0 -k) | L ′ ) = length(E (k) | L /E (k-1) | L ) + length(E (k-1) (-k[t i0 ])| L /E (k) (-(k + 1)[t i0 ])| L ) = length(E (k) | L /E (k) (-[0])| L ) = 1.
Then we obtain that

Stab w (L) = deg(E) -2 deg(L) + ni 0 k=1 w i0,k (1 -2 • length(E (k) | L /E (k-1) | L ))) + i∈I\{i0} k ǫ i,k (L)w i,k = deg(E ′ i0 ) + n i0 -2 deg(L ′ ) + n i0 - ni 0 k=1 length(E (k) | L /E (k-1) | L )) + ni 0 k=1 w i0,k 2 • length((E ′ i0 ) (ni 0 -k+1) | L ′ /(E ′ i0 ) (ni 0 -k) | L ′ ) -1 + i∈I\{i0} k ǫ i,k (L)w i,k = deg(E ′ i0 ) -2 deg(L ′ ) + ni 0 k ′ =1 (1 -2 • length((E ′ i0 ) (k ′ ) | L ′ /(E ′ i0 ) (k ′ -1) | L ′ )) + ni 0 k ′ =1 (-w i0,ni 0 -k ′ +1 )(1 -2 • length((E ′ i0 ) (k ′ ) | L ′ /(E ′ i0 ) (k ′ -1) | L ′ )) + i∈I\{i0} k ǫ i,k (L)w i,k = Stab w ′ (L ′ ).
Corollary 44. If (E, {l i,• } i∈I ) is tame and undecomposable, then elm - ni 0 [ti 0 ] (E, {l i,• } i∈I ) is also tame and undecomposable.

Proof. Since (E, {l i,• } i∈I ) is tame and undecomposable, (E, {l i,• } i∈I ) is stable for some weights w (Theorem 1). By Proposition 43, the elementary transformation elm - ni 0 [ti 0 ] (E, {l i,• } i∈I ) stable for some weights w ′ . By Theorem 1, elm - ni 0 [ti 0 ] (E, {l i,• } i∈I ) is also tame and undecomposable. 6. Geometry of moduli spaces and weak del Pezzo surfaces Let Bun(D, 1) be the moduli space of tame and undecomposable refined parabolic bundles on (P 1 , D) whose degree are one. To construct a good moduli space, we need the stability condition with the w-stability index (or, equivalently, with the α-parabolic degree). Then for weights w, we may construct the coarse moduli space Bun w (D, 1) of w-semistable refined parabolic bundles on (P 1 , D) whose degrees are one as a scheme. Moreover, this coarse moduli scheme is projective [START_REF] Yokogawa | Compactification of moduli of parabolic sheaves and moduli of parabolic Higgs sheaves[END_REF]Corollary 5.13]. Remark that the open subset of w-stable parabolic bundles is smooth [START_REF] Yokogawa | Infinitesimal deformation of parabolic Higgs sheaves[END_REF]. By Theorem A, for a tame and undecomposable refined parabolic bundle, there exist weights w such that this refined parabolic bundle is w-stable. So Bun(D, 1) is covered by the projective schemes Bun w (D, 1) when w runs.

The purpose of this section is to give a proof of Theorem B. That is, we will give detail description of the moduli spaces Bun w (D, 1) with democratic weights for n = 5. If we take generic weights w, then w-semistable = w-stable. So Bun w (D, 1) is a smooth projective scheme in this case. We will consider Bun w (D, 1) only for generic democratic weights w. If n = 5, then the dimension of Bun w (D, 1) is two. We will give description of these two-dimensional smooth projective schemes, and see that these schemes are smooth projective surfaces. We investigate the change of the smooth projective surfaces when the democratic weights vary. Moreover we will investigate the automorphism groups of the smooth projective surfaces when all the democratic weights are 1/2. Remark that, in this case, the elementary transformations for refined parabolic bundles give automorphisms on Bun 1 2 (D, 1) by Proposition 43.

Any two non trivial extensions define isomorphic parabolic bundles if, and only if, the corresponding cocycles are proportional. So the moduli space of extension is parametrized by PH 0 (P 1 , O(-1) ⊗ Ω 1 P 1 (D)) * ∼ = P n-3 . 6.1. Moduli of stable refined parabolic bundles with democratic weights for n = 5 (Proof of Theorem B). We assume that n = 5. Let D be one of the effective divisors in the following list:

D 2111 := 2[t 1 ] + [t 2 ] + [t 3 ] + [t 4 ], D 221 := 2[t 1 ] + 2[t 2 ] + [t 3 ], D 311 := 3[t 1 ] + [t 2 ] + [t 3 ], D 32 := 3[t 1 ] + 2[t 2 ], D 41 := 4[t 1 ] + [t 2 ], D 5 := 5[t 1 ].
Our purpose here is to describe moduli spaces of stable refined parabolic bundles with democratic weights w = (w 1 , . . . , w ν ) for n = deg(D) = 5. Here the democratic weights means that w i = (w i,ni , . . . , w i,2 , w i,1 ) = (w, . . . , w, w)

for 0 < w < 1. If 1 5 < w < 1 3 , then this moduli space is P 2 (see Proposition 47). We would like to describe the moduli spaces for another democratic weights w. If w is moved and w crosses walls, then some "special bundles" (which are stable for 1 5 < w < 1 3 ) become unstable and some "special bundles" (which are unstable for 1 5 < w < 1 3 ) become stable. Here, the "special bundles" are defined in the appendix below. To describe the moduli spaces, we consider such a wall-crossing.

Definition 48. We define special bundles and the types of special bundles as in the appendix below. There are 6 types: type A, type B,. . ., type F. We say a tame and undecomposable refined parabolic bundle is generic if this refined parabolic bundle is not a special bundle.

We may check that the following claims:

• If a tame and undecomposable refined parabolic bundle is generic, then this refined parabolic bundle is stable for any w ( 1 5 < w < 1); . So there are loci on P 2 corresponding to special bundles with type A, type B, and type C. The locus corresponding special bundles with type A is a conic all intersections. Below, the curve C is identified with the base curve of the parabolic bundle for simplicity by (6.1). Note that the one thick line is (-2)-curves. In the previous section, we had seen that some elementary transformations of refined parabolic bundles induce automorphisms on the moduli space Bun w (D 2111 , 1) ∼ = wdP

L 2[t1] L [t1],[t4] L [t1],[t3] L [t1],[t2] L [t2],[t3] L [t2],[t4] L [t3],[t4] P [t1] P [t2] P [t3] P [t4] C E (1) [t1] E (0) [t1] L [t1],[t2] L [t1],[t3] L [t1],[t4] L 2[t1] L [t2] L [t3] L [t4] C L [t3],[t4] L [t2],[t4] L [t2],[t3] L [t2],[t4] L [t2],[t3]
A1 = elm [t2]+[t3] , elm [t3]+[t4] , elm 2[t1] .
Proof. These three elementary transformations act as follows on the negative curves:

• all of them stabilize the unique (-2)-curve E

[t1] (and so does any automorphism), and therefore permute the 4 intersecting (-1)-curves

E (0) [t1] , L [t1],[t2] , L [t1],[t3] , L [t1],[t4] ;
• the action on these 4 curves is by double-transposition, since the crossratio of their interection point with the (-2)-curve E

[t1] should be preserved and is generic (coinciding with the cross-ratio of (C,

[t 1 ] + [t 2 ] + [t 3 ] + [t 4 ])); • elm [t2]+[t3] permutes L [t1],[t2] with L [t1],[t3] ; • elm [t3]+[t4] permutes L [t1],[t3] with L [t1],[t4] ; • elm 2[t1] stabilizes all 4 curves E (0) [t1] , L [t1],[t2] , L [t1],[t3] , L [t1],[t4] , but permutes C and L 2[t1] .
One easily check these properties by studying the action of elementary transformations on parabolic structure and special line bundles. One also deduce that the 3 elementary transformations (which commute) indeed generate a order 8 group of automorphisms isomorphic to Z/2 × Z/2 × Z/2.

It remains to check that any automorphism φ belongs to that 8 order group. One can assume, after composition by an element of the 4 order group elm

[t2]+[t3] , elm [t3]+[t4] , that φ stabilizes all 4 curves E (0) [t1] , L [t1],[t2] , L [t1],[t3] , L [t1],[t4] . Since E (0)
[t1] intersects only 3 negative curves, namely C and L 2[t1] and the (-2)-curve E

[t1] , it has to permute the two first ones. By composing with elm 2[t1] , we can assume that φ also stabilizes each of the two curves C and L 2[t1] . Then, the intersection combinatorics shows that φ eventually stabilizes all negative curves. Therefore, we can blow-down back to P 2 and get a linear automorphism φ which fixes the 4 points supporting the divisor on the conic. There exists an one-to-one correspondence between linear automorphisms on P 2 which preserve the conic and automorphisms on the conic. Therefore φ is the identity, φ as well, obviously belonging to the 8 order group.

If D has special configuration, then we might have non trivial automorphism ϕ ∈ Aut(C, D 2111 ). For the non trivial automorphism ϕ, we can pull-back a refined parabolic bundle by ϕ. Since we impose that the weights are democratic, the stability condition of this pull-back is the same as the stability condition of the refined parabolic bundle. So, in that case, the automorphism ϕ induces an extra automorphism for the weak del Pezzo surface. Now we set D = D 221 . The left-hand-side of Figure 2 is the projective plane P 2 with the locus of special bundles. The right-hand-side of Figure 2 is the graph of negative curves on the weak del Pezzo surface obtained by blowing-up of the left-hand-side. This weak del Pezzo surface wdP is just the moduli space Bun 1 2 (D 221 , 1). Note that C and L [t3] intersect each other, although we don't see on the picture. We take the automorphism ϕ ∈ Aut(C, D 221 ) permuting t 1 and t 2 , fixing t 3 . Let ϕ * be the automorphism of wdP (4) 2A1 induced by the pull-backs of refined parabolic bundles by ϕ. Proposition 50. The automorphism group of the weak del Pezzo surface is Z/2 ⋉ (Z/2 × Z/2), generated by: Aut wdP

P [t1] P [t2] P [t3] C L [t1],[t2] L [t1],[t3] L [t2],[t3] L 2[t1] L 2[t2] E (1) [t1] E (1) [t2] L [t3] C L [t1],[t2] L [t1],[t3] L [t2],[t3] L 2[t1] L 2[t2] E (0) [t1] E (0) [t2]
(4) 2A1 = ϕ * , elm 2[t1] , elm 2[t2] .
Proof. It is similar to the previous one. One can check:

• any automorphism permutes the two (-2)-curves E

[t1] and E

[t2] , and therefore stabilize the unique (-1)-curve intersecting them, namely L [t1],[t2] ;

• ϕ permutes non trivially E

[t1] and E

(1) [t2] ; • elm 2[t1] stabilizes E (1) [t1] , E (0) [t1] , L [t1],[t3] , E (1)
[t2] but permutes E

[t2] , E

[t2] and L [t2],[t3] , but permutes E (0)

[t1] and L [t1],[t3] non trivially. Given an automorphism φ, one can assume, after composition by an element of the 8 order group of the statement, that is stabilizes all 7 curves E (i)

[tj ] , L [ti],[tj] ; but combinatorics imply that φ stabilizes all negative curves: its blow-down φ on P 2 has to preserve the conic and fix support of D 221 . It is the identity. Now we set D = D 311 . The left-hand-side of Figure 3 is the projective plane P 2 with the locus of special bundles. The right-hand-side of Figure 3 is the graph of negative curves on the weak del Pezzo surface obtained by blowing-up of the left-hand-side. This weak del Pezzo surface wdP (4) A2 is just the moduli space Bun 1 2 (D 311 , 1). Note that the two thick lines are (-2)-curves.

P [t1] P [t2] P [t3] C L [t2],[t3] L [t1],[t3] L [t1],[t2] L 2[t1] E (2) [t1]
E

L We take the automorphism ϕ ∈ Aut(C, D 311 ) permuting t 2 and t 3 , fixing t 1 . Let ϕ * be the automorphism of wdP Given an automorphism φ, one can assume, after composition by an element of the 8 order group of the statement, that is stabilizes all 6 curves E (i)

[t2] L [t3] C L [t2],[t3] L [t1],[t3] L [t1],[t2] L 2[t1] E (0) [t1]
[t1] , L [t1],[tj] and L 2[t1] ; but combinatorics imply that φ stabilizes all negative curves: its blow-down φ on P 2 has to preserve the conic and fix support of D 311 . It is the identity. Now we set D = D 32 . The left-hand-side of Figure 4 is the projective plane P 2 with the locus of special bundles. The right-hand-side of Figure 4 is the graph of negative curves on the weak del Pezzo surface obtained by blowing-up of the left-hand-side. This weak del Pezzo surface wdP (4) A2+A1 is just the moduli space Bun 1 2 (D 32 , 1). Note that the three thick lines are (-2)-curves. Given an automorphism φ, one can assume, after composition by elm 2[t2] that is stabilizes all negative curves and blow-down as a linear automorphism of P 2 stabilizing the conic C and fixing the two points P [t1] and P [t2] . It is determined by its action on the conic; there is an element ϕ ∈ Aut(C, D 32 ) inducing the same action on the conic: φ coincides with the natural action of ϕ on refined parabolic bundles. So Aut wdP Proof. One can check:

P [t1] P [t2] C L [t1],[t2] L 2[t1] L 2[t2] L [t1],[t2] E (2) [t1] E (1) [t1] E (0) [t1] C E (1) [t2] E (0) [t2] L 2[t1] L 2[t2]
• any automorphism stabilizes E Proof. By intersection combinatorics, any automorphism φ must stabilize all negative curves, and therefore blow-down as a linear automorphism of P 2 stabilizing the conic C and fixing the point P [t1] .

Appendix: List of special bundles

In this appendix, we will give a list of special bundles for n = 5. First, we give a list of standard tableaus of refined parabolic structures, which appear in the cases where n = 5. If the multiplicity of a point is 2, then the standard tableau of a refined parabolic structure at this point has two possibility:

•

  Painlevé equation P V I when D = [t 1 ] + [t 2 ] + [t 3 ] + [t 4 ]; • Garnier systems when deg(D) ≥ 5 and D is reduced; • Painlevé equations P V , P IV , P III(D (1) 6 ) and P II when D = [t 1 ] + [t 2 ] + 2[t 3 ], D = [t 1 ] + 3[t 2 ], D = 2[t 1 ] + 2[t 2 ], and D = 4[t 1 ], respectively; and

  A 3 and A 4 when D = D 2111 , D 221 , D 311 , D 32 , D 41 , and D 5 , respectively.

  the condition (a) implies that a Λ-connection (E, ∇) has unramified irregular singular point at t i . The degree of E is d by (b). The following is [4, Lemma 4.6]: Proposition 6. Let Λ satisfying (a),(b),(c),(d) as above. Then, any Λ-connection (E, ∇) is irreducible.

  then we have by Fuchs' relation for ∇ 0 and assumption (b) for Λ, we have Res(ω α ) = Res(tr(∇ α )) -Res(tr(∇ 0 )) = 0.

dx x 2 and a 1 dx

 1 (x-1) 2 respectively. Flatness condition writes c 0 a 0 + c 1 a 1 = 0 which determines the parabolic bundle uniquely. Now we consider the case where n = deg(D) = 4 and deg(E) is even. We give counterexamples of the implication [(ii) ⇒ (i)] and the implication [(iii) ⇒ (ii)]. Proposition 12. Let n = deg(D) = 4, Λ as above, (E, l) a parabolic bundle, deg(E) any even number.

  the case where n = deg(D) = 5 and deg(E) is odd. Lemma 13. Let n = deg(D) = 5 and (E, l) a parabolic bundle with odd degree. Assume that (E, l) is undecomposable and admissible

  Now we give counterexamples when n = deg(D) = 5 and deg(E) is odd. Proposition 14. Let n = deg(D) = 5, Λ as above, and (E, l) a parabolic bundle with odd degree.

  and parabolic structure writes as in (ii) of Lemma 13 where [c 1 : c 2 : c 3 ] lies along some line determined by Λ; • D = 4[t 1 ] + [t 2 ] and parabolic structure writes as in (iii) of Lemma 13 where [c 1 : c 2 ] is a point determined by Λ; • D = 5[t 1 ] and parabolic structure writes as in (iv) of Lemma 13 where [c 1 : c 2 : c 3 ] lies along some line determined by Λ.

(3. 1 )

 1 a 1 (λ) := #{j | j ∈ {1, 2, . . . , n}, λ j = 1} and a 2 (λ) := #{j | j ∈ {1, 2, . . . , n}, λ j = 2}. Proposition 17. If a tuple of integers λ = (λ n , λ n-1 , . . . , λ 1 ) satisfies the condition

.

  So there exists the corresponding Young diagram to the type λ. The number of columns of the Young diagram is 1 or 2. Lemma 18. Let l be an O n[t] -submodule of E| n[t] with length(l) = k. Let λ = (λ n , λ n-1 , . . . , λ 1 ) be the type of l. If the number of columns of the corresponding Young diagram is 1, then the O n[t]submodule l is generated by one element of E| n[t] . If the number of columns of the corresponding Young diagram is 2, then the O n[t] -submodule l is generated by two elements of E| n[t] .

( 4 ) 2 → 4 ) 2 .

 4242 F 2 this P 1 -bundle. By the gluing the families, we have a family of filtrations l n ⊃ ln-1 ⊃ ln-2 ⊃ ln-3 parametrized by F (Points on a Zariski open subset of F (4) 2 parametrize filtrations whose sequences of types are T 4 . We may check that F (4) 2

T ⊃ 0

 0 with the fixed l n parametrized by F (n) ln,T such that on a Zariski open subset C a2 ⊂ F (n) ln,T the restriction l•,T | C a 2 gives a bijection

  and define ln-k,T by the gluing of li n-k .

  not empty, and •deg(E) + 2 deg(L) + 1 ≤ i∈I + L N i (L) for any line subbundles L such that deg(E) ≤ 2 deg(L). If E has a decomposition E ∼ = O(d 1 ) ⊕ O(d 2 ) where d 1 < d 2 , then there exists a unique line subbundle L such that deg(E) ≤ 2 deg(L). This line subbundle is the destabilizing bundle O(d 2 ).

  Then the vector bundle O(d 1 ) ⊕ O(d 2 ) has the following automorphism: where c ∈ C * and G = G(x) be a polynomial of degree less than or equal to d 2d 1 . By (4.6) and (4.7), there exists an automorphism of O

  L1) ǫ i2,k (L 1 ). For any line subbundles L ⊂ E with deg(L) = d and L = L 1 , we have the following inequality:

Figure 1 .

 1 Figure 1. P 2 and wdP (4) A1

( 4 ) 4 )

 44 A1 . Now we reconstruct all automorphisms of the weak del Pezzo surface wdP (A1 via these elementary transformations. Let Aut(C, D 2111 ) be the automorphism of C sending D 2111 to itself. Remark that any automorphisms in Aut(C, D 2111 ) fix the point t 1 . If the points t 1 , t

Figure 2 .

 2 Figure 2. P 2 and wdP (4) 2A1

  and L [t2],[t3] non trivially; • in a similar way, elm 2[t2] stabilizes E

Figure 3 .

 3 Figure 3. P 2 and wdP (4) A2

( 4 ) 4 )

 44 A2 induced by the pull-backs of refined parabolic bundles by ϕ. Proposition 51. The automorphism group of the weak del Pezzo surface is Z/2 ⋉ (Z/2 × Z/2), generated by: Aut wdP(A2 = ϕ * , elm 3[t1]+[t2] , elm [t2]+[t3] . Proof. Itis similar to the previous ones. One can check: • any automorphism permutes the two (-2)-curves E ; • elm 3[t1]+[t2] permutes non trivially E and L 2[t1] , but permutes non trivially the two lines L [t1],[t2] and L [t1],[t3] ; • elm [t2]+[t3] stabilizes E , but permutes non trivially the two lines E (0) [t1] and L 2[t1] .

Figure 4 .

 4 Figure 4. P 2 and wdP (4) A2+A1Let Aut(C, D 32 ) be the group of automorphisms fixing t 1 and t 2 . We have an isomorphism Aut(C, D 32 ) ∼ = C * .Proposition 52. The automorphism group of the weak del Pezzo surface is Z/2 × C * , generated by: Aut wdP(4) A1+A2 = elm 2[t1] , Aut(C, D 32 ) .Proof. It is similar to the previous ones. One can check:• any automorphism stabilizes E (i) [t1] , i = 1, 2, E(i)[t2] , i = 0, 1, and L [t1],[t2] by intersection combinatorics between (-2) and (-1)-curves;• elm 2[t2] permutes L 2[t1] with E (0)[t1] , and L 2[t2] with C. Given an automorphism φ, one can assume, after composition by elm 2[t2] that is stabilizes all negative curves and blow-down as a linear automorphism of P 2 stabilizing the conic C and fixing the two points P [t1] and P [t2] . It is determined by its action on the conic; there is an element ϕ ∈ Aut(C, D 32 ) inducing the same action on the conic: φ coincides with the natural action of ϕ on refined parabolic bundles. So Aut wdP

( 4 )

 4 A1+A2 is generated by elm 2[t1] and Aut(C, D 32 ). Since ϕ ∈ Aut(C, D 32 ) fixes t 1 and t 2 , we may check thatelm 2[t1] (ϕ * E, ϕ * l) = ϕ * elm 2[t1] (E, l)for any (E, l) ∈ Bun 1 2 (D 32 , 1). That is, the automorphisms ϕ * and elm 2[t1] commute. So the automorphism group of the weak del Pezzo surface is isomorphic to Z/2 × C * . Now we set D = D 41 . The left-hand-side of Figure5is the projective plane P 2 with the locus of special bundles. The right-hand-side of Figure5is the graph of negative curves on the weak del Pezzo surface obtained by blowing-up of the left-hand-side. This weak del Pezzo surface wdP

Figure 5 . 2 . 4 )

 524 Figure 5. P 2 and wdP (4) A3 Let Aut(C, D 41 ) be the group of automorphisms fixing t 1 and t 2 . Proposition 53. The automorphism group of the weak del Pezzo surface is Z/2 × C * , generated by: Aut wdP (4) A3 = elm 4[t1] , Aut(C, D 41 ) .

4 )

 4 and L 2[t1] ;• elm4[t1] permutes E . Given an automorphism φ, one can assume, after composition by elm4[t1] that is stabilizes all negative curves, and therefore blow-down as a linear automorphism of P 2 stabilizing the conic C and fixing the two points P [t1] and P [t2] . So Aut wdP (A3 is generated by elm 4[t1] and Aut(C, D 41 ). Since ϕ ∈ Aut(C, D 41 ) fixes t 1 and t 2 , the automorphisms elm 4[t1] and ϕ * commute. Now we set D = D 5 . The left-hand-side of Figure 6 is the projective plane P 2 with the locus of special bundles. The right-hand-side of Figure 6 is the graph of negative curves on the weak del Pezzo surface obtained by blowing-up of the left-hand-side. This weak del Pezzo surface wdP

Figure 6 .

 6 Figure 6. P 2 and wdP (4) A4

TII : ( 2 )m 1 [[= 1

 211 ⊃ (1). If the multiplicity of a point is 3, then the standard tableau of a refined parabolic structure at this point has three possibility: 7.1. Case D = D 2111 . Now we describe special refined parabolic bundles which are undecomposable and tame when D = D 2111 . [Type A]. E = O ⊕ O(1) and m t1,O(1) = • • • = m t4,O(1) = 0. T l1,• there exists a subundle O(-1) ⊂ E as below: t1,O(-1) = 2, m t2,O(-1) = m t3,O(-1) = m t4,O(-1) = Type B]. E = O ⊕ O(1) and m t1,O(1) = • • • = m t4,O(1) = 0.T l1,• there exists a subundle O ⊂ E as below:L [ti],[tj ] T (2) I m t1,O = 2, m t k ,O = 1 where k ∈ {2, 3, 4} \ {i, j}. L [t1],[ti] T (2) I m t1,O = m tj ,O = m t k ,O = 1 where {j, k} = {2, 3, 4} \ {i}. L 2[t1] T (2) I m t2,O = m t3,O = m t4,O = 1 for i, j ∈ {2, 3, 4} with i = j. [Type C]. P [t1] := C ∩ L 2[t1] , P [ti] := C ∩ Type D]. E = O ⊕ O(1).T l1,• there exists a subundle O(1) ⊂ E as below:for i, j ∈ {1, 2, 3, 4}.[Type E]. E = O ⊕ O(1). T l1,• there exists a subundle O(1) ⊂ E as below: II m t1,O(1) = m t i ′ ,O(1) = 1. for i, j ∈ {1, 2, 3, 4} with i = j and for i ′ ∈ {2, 3, 4}. [Type F]. E = O(-1) ⊕ O(2) and T lt 1 ,• : (1, 1) ⊃ (1).

7. 2 .mmm= 1 m 1 [m= 1 [= 2 [ 7 . 6 .= 5 [

 21112765 Case D = D 221 . Now we describe special refined parabolic bundles which are undecomposable and tame when D = D 221 . [Type A]. E = O ⊕ O(1) and mt1,O(1) = m t2,O(1) = m t3,O(1) = 0. T l1,• T l2,• there exists a subundle O(-1) ⊂ E as below: t1,O(-1) = m t2,O(-1) = 2, m t3,O(-1) = 1. [Type B]. E = O ⊕ O(1) and m t1,O(1) = m t2,O(1) = m t3,O(1) = 0. T l1,• T l2,•there exists a subundle O ⊂ E as below:L [t1],[t2] T t1,O = m t2,O = m t3,O = 1. L [ti],[t3] T m t i ′ ,O = 2, m t3,O = 1 where i ′ ∈ {1, 2} \ {i}. L 2[ti] T t i ′ ,O = 2, m t3,O = 1 where i ′ ∈ {1, 2} \ {i}.for i ∈ {1, 2}.[Type C].P [t1] := C ∩ L 2[t1] , P [t2] := C ∩ L 2[t2] , P [t3] := C ∩ L [t1],[t3] ∩ L [t2],[t3] . [Type D]. E = O ⊕ O(1).T l1,• T l2,• there exists a subundle O(1) ⊂ E as below:for i ∈ {1, 2, 3}.[Type E]. E = O ⊕ O(1). T l1,• there exists a subundle O(-1) ⊂ E as below: t1,O(-1) = 4, m t2,O(-1) = Type B]. E = O ⊕ O(1) and m t1,O(1) = m t2,O(1) = 0.T l1,• there exists a subundle O ⊂ E as below:L [t1],[t2] T t1,O = 2, m t2,O = 1. [Type C]. P [t1] := C ∩ L 2[t1] , P [t2] := (C ∩ L [t1],[t2] ) \ P [t1].[Type D]. E = O ⊕ O(1). T l1,• there exists a subundle O(1) ⊂ E as below: Type E]. E = O ⊕ O(1). T l1,• there exists a subundle O(1) ⊂ E as below: IV m t1,O(1) = m t2,O(1) = 1. and T l1,• there exists a subundle O(1) ⊂ E as below: Type F]. E = O(-1) ⊕ O(2) and T l1,• : (1, 1, 1) ⊃ (1, 1) ⊃ (1). Case D = D 5 . Now we describe special refined parabolic bundles which are undecomposable and tame when D = D 5 . [Type A]. E = O ⊕ O(1) and m t1,O(1) = 0. T l1,• there exists a subundle O(-1) ⊂ E as below: Type B]. E = O ⊕ O(1) and m t1,O(1) = 0. T l1,• there exists a subundle O ⊂ E as below:

  • Id E , and by linearity, it is therefore enough to test the vanishing of {ω αβ } for A = Id E .

	Fix any global meromorphic connection ∇ 0 on E; for instance, one can pull-back the trivial
	connection d from any birational trivialization E	C 2 . It obviously satisfies Fuchs' relation
	Res ∇ 0 = -deg(E). Now consider the following meromorphic resolution of the cocycle

Table 1 .

 1 Special bundles with type E are unstable for any democratic weights w. These can be summarized as Table1. Table of stability of special bundles

	• If 1 5 < w < 1 3 , then special bundles with type A, type B, and type C are stable. On the other hand, special bundles with type D and type F are unstable;
	• If 1 3 < w < 3 5 , then special bundles with type C become unstable. Instead of it, special bundles with type D become stable;
	• If 3 5 < w < 1, then special bundles with type A become unstable. Instead of it, special bundles with type F become stable;
	• type A	type B	type C	type D	type E	type F
	0 < w < 1 5 1 5 < w < 1 3 1 3 < w < 3 5 3 5 < w < 1 unstable unstable unstable unstable unstable unstable unstable stable stable stable unstable unstable unstable stable stable unstable stable unstable unstable stable unstable stable unstable stable
	When 1 5 < w < 1 3 , Bun w (D, 1) ∼ = P 2

  2 , t 3 , and t 4 have a generic cross-ratio, then the non trivial automorphisms in Aut(C, [t 1 ]+[t 2 ]+[t 3 ]+[t 4 ]) act on the 4 points by double-transposition: That is, (t 1 ↔ t 2 , t 3 ↔ t 4 ), (t 1 ↔ t 3 , t 2 ↔ t 4 ), and (t 1 ↔ t 4 , t 2 ↔ t 3 ). So Aut(C, D 2111 ) = id . Proposition 49. If Aut(C, D 2111 ) = id , then the automorphism group of the weak del Pezzo surface is Z/2 × Z/2 × Z/2, generated by elementary transformations:

	Aut wdP	(4)

The first author is supported by JSPS KAKENHI: Grant Numbers JP17H06127 and JP19K14506. The second author is supported by CNRS, ANR-16-CE40-0008 project "Foliage" and Henri Lebesgue Center, program ANR-11-LABX-0020-0. The third author is supported by JSPS KAKENHI: Grant Number JP17H06127, JP22H00094. 1

Definition 39. We fix i 0 where i 0 ∈ I. We define the elementary transformation of (E, {l i,• } i∈I ) at n i0 [t i0 ] by elm - i0 (E, {l i,• } i∈I ) = (E ′ i0 , {l i,• } i∈I\{i0} ∪ {l ′ i0,• }). The degree of E ′ i0 is dn i0 . 5.4. Some properties of elementary transformations for refined parabolic bundles. We take i ∈ I. Let l i,• : l i,ni ⊃ • • • ⊃ l i,1 ⊃ 0 be a refined parabolic structure at n i [t i ] and l ′ i,• be its transformation. For l i,• and l ′ i,• , we can define standard tableaus T and T ′ , respectively (see Definition 19). We will consider the relation between these standard tableaus T and T ′ . Let λ (k) i be the Young diagram which is the type of l i,k . We define integers a k 1 and a k 2 so that λ

2 ) for k = 1, 2, . . . , n i . We set

and X 2 (k) := a ni 2a k 2 . Definition 40. We say a refined parabolic structure l i,• is generic if

• a ni 2 = 0, or • a ni 2 = 0 and we can take generators v 1 , v 2 ∈ l i,ni such that length( v 1 ) = a ni 1 + a ni 2 , length( v 2 ) = a ni 2 , and l i,• satisfies

for any k with a k+1 2 a k 2 > 0. Now we consider examples of generic l i,• . Set λ

). So a ni 1 = n i -2 and a ni 2 = 1. Let T k be the standard tableau defined in (3.4) for k = 1, 2, . . . , n -1. We fix k 0 (1 ≤ k 0 ≤ n -1). If the standard tableau of a refined parabolic bundle is T k0 , then a n-k0+1 2 a n-k0 2 = 1 > 0 and a k+1 2 a k 2 = 0 when k = nk 0 . We take elements v 1 , v 2 ∈ E| n[t] such that length( v 1 ) = n i -1, and length( v 2 ) = 1, and l i,ni = v 1 , v 2 O n [t] . In particular, f ni-1 i v 1 = f i v 2 = 0. We can describe a refined parabolic structure as follows:

where α ∈ C. This refined parabolic structure is generic if, and only if, α = 0.

To describe the standard tableau of the transformation l ′ i,• , first, we define integers b ni , b ni-1 , . . . , b 1 by b ni = a ni 1 and b k = min{b k+1 , a k 1 } for k = n i -1, . . . , 2, 1. Second, we modify X 1 (k) and X 2 (k) by using b k as follows:

Proof. When a ni 2 = 0, it is clear. So we consider the case where a ni 2 = 0. We take generators v 1 , v 2 of l i,ni as in the definition of the genericness of l i,• . We can take

We assume that l i,k has the description (5.3) for the fixed k. Since X b 2 (k) ≤ X 2 (k) and F (0) = 0, we have an inequality

First, we consider the case

for the fixed k. In this case, the submodule l i,k-1 has the following descriptions:

, there exists

Here remark that

Second, we consider the case a k 1 = a k-1

1

-1 and a k 2 = a k-1 2 + 1 for the fixed k. In this case, the submodule l i,k-1 has the following descriptions:

since v 1 and v 2 satisfy the condition of the genericness of l i,• . Here remark that

Finally, we have the claim that we may take F (f i ) ∈ O ni[ti] (where F (0) = 0) so that (5.3).

We take ṽ1 , ṽ2

respectively. Then we have

. We may check the following equalities

If a ni 2 = 0 or a ni 2 = 1, then we have a simple description of the standard tableau of the transformation l ′ i,• . If a ni 2 = 0, the standard tableau of l i,• is unique. We denote by T 0 this standard tableau. Let T ′ be the standard tableau of the transformation

is also an ordinary parabolic bundle. When a ni 2 = 1, description of the standard tableau of the transformation l ′ i,• is as follows:

Corollary 42. We assume that λ (n) = (1 n-2 , 2 1 ). Let T k be the standard tableau defined in (3.4) for k = 1, 2, . . . , n -1. Let l i,• be a generic refined parabolic structure with the standard tableau T k . Let l ′ i,• be the transformation (5.2) of l i,• and T ′ be the standard tableau of l ′ i,• . Then T ′ = T n-k . Proof. By direct computation, we may check that

By Proposition 41, we have the claim. Now, we discuss a relation between the elementary transformations and the stability condition.

A refined parabolic bundle (E, l) of degree 1 is w-stable if, and only if, (E, l) satisfies the following conditions:

) is an undecomposable parabolic bundle (in particular, l i is free for any i ∈ I);

• l is in general position.

Proof. We assume that

We have the following inequality:

So this refined parabolic bundle is not stable. We assume that E = O ⊕ O(1) and there is a refined parabolic structure which intersects O(1). Then

So this refined parabolic bundle is not stable. This implies that l is a (ordinary) parabolic structure and l is in general position. We assume that E = O ⊕ O(1), the parabolic bundle is decomposable, and there is no parabolic structure which intersects O(1). Then there is a subbundle O ⊂ E such that any parabolic structures are contained in O. Then ,(E,l) is an undecomposable parabolic bundle, and l is in general position.

We assume that (O ⊕ O(1), l) is undecomposable and l is in generic position. Since there is no refined parabolic structure which intersects O(1), we have

Proposition 47. We set w i,k = w (for any

We have an isomorphism Bun w (D, 1) ∼ = P n-3 .

Proof. By Proposition 46, a w-stable refined parabolic bundle (E, l) of degree 1 is a non trivial extension 0 → (O(1), ∅) → (E, l) → (O, l) → 0. The obstruction to split the extension is measured by an element of

on P 2 . The loci corresponding special bundles with type B are lines on P 2 . The loci corresponding special bundles with type C are intersections the conic and the lines. We may define an embedding (6.1)

such that the image is the conic corresponding to special bundles with type A and the images of the points t 1 , t 2 , . . . , t ν are points corresponding to special bundles with type C. We may describe the configuration of the loci of special bundles of type A, type B, and type C as follows.

• When D = D 2111 , the configuration of the loci of special bundles is as follows:

, and the line L [ti],[tj] is intersect to the conic C at P [ti] and P [tj ] for i, j ∈ {1, 2, 3, 4} with i = j. • When D = D 221 , the configuration of the loci of special bundles is as follows:

, and the line

is intersect to the conic C at P [ti] and P [tj ] for i, j ∈ {1, 2, 3} with i = j. • When D = D 311 , the configuration of the loci of special bundles is as follows:

is tangent to the conic C at P [t1] , and the line L [ti],[tj] is intersect to the conic C at P [ti] and P [tj ] for i, j ∈ {1, 2, 3} with i = j. • When D = D 32 , the configuration of the loci of special bundles is as follows: • When D = D 5 , the configuration of the loci of special bundles is as follows:

Now, we will describe the moduli space Bun w (D, 1) with 1 3 < w < 3 5 . Let w ′ be democratic weights with 1 5 < w ′ < 1 3 . We identify Bun w ′ (D, 1) with P 2 . We consider the sequence of blowing ups

of Bun w ′ (D, 1) at points which have the following position:

Here P ′ ≻ C P means P ′ is the intersection of the exceptional divisor of the blowing up at P and the strict transformation of the conic C. Here P is a point on C or a strict transformation of C. We denote by wdP (n) X the weak del Pezzo surface of degree n whose configuration of (-2)-curves is X. We have that X 5 (D 2111 ) = wdP

Let E (k)

[ti] (k = 0, 1, 2, . . . , n i -1) be the locus of special bundles with type D on the moduli space Bun w (D, 1). By using Corollary 27, E

[ti] (k = 1, 2, . . . , n i -1) compose a chain of (n i -1) projective lines. We may check that E (k)

[ti] is the exceptional divisor of the blowing up at P (k) [ti] (k = 0, 1, 2, . . . , n i -1). Here we set P

. Now, we will describe the moduli space Bun w (D, 1) with 3 5 < w < 1. We may check that the strict transformation of the conic C on X 5 (D) is a (-1)-curve. We contract this (-1)-curve on X 5 (D):

π ′ 5 : X 5 (D) -→ X ′ 5 (D). Then we have weak del Pezzo surfaces of degree 5:

A4 . We have that

Let P C be the locus of special bundles with type F on the moduli space Bun w (D, 1). We may check that P C = π ′ 5 (C). These can be summarized as Table 2. Now we check the assertion (ii) in Theorem B. Let Π -2 (D) be the effective divisors consisting of all (-2)-curves on Bun w (D, 1) with 1 3 < w < 3 5 . We have that

Table 2. Table of moduli spaces

).

By the list in the appendix (Section 7), we may check the following claim: If (E, {l i,• } i ) be a refined parabolic bundle corresponding to a point on Π -2 (D), then there exists i (1 ≤ i ≤ ν) such that l i,ni is not free, that is, l i,• is not a parabolic structure for this i.

Next we check the assertion (iii) in Theorem B. Let (E, {l i,• } i ) be a refined parabolic bundle corresponding to a point on Bun w (D, 1) with 1 5 < w < 1 3 . By Proposition 46, (E, {l i,• } i ) is an ordinary parabolic bundle. By Lemma 13 (or Proposition 35), we have that (E, {l i,ni } i ) is a simple parabolic bundle. Let Π -2 (D) ′ be the effective divisor on Bun w (D, 1) with 3 5 < w < 1 induced by Π -2 (D). Let (E, {l i,• } i ) be a refined parabolic bundle corresponding to a point on Bun w (D, 1) \ Π -2 (D) with 1 3 < w < 3 5 , or Bun w (D, 1) \ Π ′ -2 (D) with 3 5 < w < 1. By the argument of the effect on the moduli space of varying the weights and the list in the appendix (Section 7), we may check that (E, {l i,• } i ) is an ordinary parabolic bundle. Moreover, by Lemma 13 (or Proposition 35), we have that (E, {l i,ni } i ) is a simple parabolic bundle. Finally, we obtain the statements in Theorem B.

6.2. Automorphisms of the moduli spaces for the democratic weights with w = 1 2 . Now we introduce some automorphisms of the moduli space Bun w (D, 1) for the democratic weights with w = 1 2 . By Proposition 43, we have the following claim: If a refined parabolic bundle (E, l) is stable for democratic weights for w (0 < w < 1), then the elementary transformation of (E, l) at t i0 is stable for the weights w ′ = (w ′ i ) i∈I , where

(1w, . . . , 1w, 1w) when i = i 0 (w, . . . , w, w) when i = i 0 .

So we have an isomorphism between moduli spaces:

and n i0 is even, we set [START_REF] Araujo | On automorphisms of moduli spaces of parabolic vector bundles[END_REF]. By Corollary 42, we have the claim that the image of (-2)-curves under the automorphism O((n i0 /2)

(where n i0 ≥ 2 and n i0 is even) is the following:

We take i 1 ∈ I \ {i 0 } so that n i0 + n i1 is even. We set

, which is an automorphism of Bun 1 2 (D, 1). Since the automorphisms elm ni 0 [ti 0 ] and elm ni 1 [ti 1 ] are commutative, the automorphism elm ni 0 [ti 0 ]+ni 1 [ti 1 ] is independent of the order of n i0 [t i0 ] and n i1 [t i1 ].

6.3.

Geometry of the weak del Pezzo surfaces from the modular point of view. Now we impose that n = 5 and w

2 for any i ∈ I. Then the moduli space Bun 1 2 (D, 1) is a weak del Pezzo surface wdP

X . Here X means the configuration of (-2)-curves on the weak del Pezzo surface. The purposes of this section is to recover the geometry of the weak del Pezzo surfaces of degree 4 from the modular point of view. First we will describe the graph of negative curves on the surface wdP (4) X for each D. Second we will reconstruct all automorphisms of the surface wdP (4) X via elementary transformations of refined parabolic bundles and automorphisms of the base curve with D. Now we set D = D 2111 . The left-hand-side of Figure 1 is the projective plane P 2 with the locus of special bundles. Here the projective plane P 2 is identified with the moduli space Bun w (D 2111 , 1) with 1 5 < w < 1 3 . The right-hand-side of Figure 1 is the graph of negative curves on the weak del Pezzo surface obtained by blowing-up of the left-hand-side. This weak del Pezzo surface wdP (4) A1 is just the moduli space Bun 1 2 (D 2111 , 1). Remark that some lines are repeated in order to see T

I . If the multiplicity of a point is 4, then the standard tableau of a refined parabolic structure at this point has six possibility:

If the multiplicity of a point is 5, then the standard tableau of a refined parabolic structure at this point has ten possibility:

For giving a list of special bundles for n = 5, now, we discuss some condition for refined parabolic structures. For a subbundle L ⊂ E, we define an integer m ti,L by

We assume that (E, l) is tame and undecomposable and deg(E) = 1. In particular, (E, l) is admissible. So we have the inequality i∈I m ti,L ≤ 4 -2 deg(L)

for any line subbundles L such that deg(E) ≤ 2 deg(L). So we have the following facts: 2), then l i,ni are free for any i.

Now we consider the cases where E = O ⊕ O(1). We impose m ti,O(1) ≤ 2 for i ∈ I. We set ǫ i,k := ǫ i,k (O(1)) for i ∈ I and k ∈ {1, 2, . . . , n i }. When we will check the tameness of refined parabolic bundles, we will calculate N i (O(1)), which is defined in (4.2). Now the tameness means that

To calculate N i (O(1)), we consider the tuple (ǫ i,ni , . . . , ǫ i,1 ). The tuple (ǫ i,ni , . . . , ǫ i,1 ) is generically determined by the pair (T li,• , m ti,O(1) ). Here T li,• is the standard tableau of the refined parabolic structure l i,• at t i . Now we will describe the correspondence between (ǫ i,ni , . . . , ǫ i,1 ) and (T li,• , m ti,O(1) ) in Table 3, Table 4, Table 5, and Table 6.

Table 4. The correspondence with

III , 2) (-, +, -) 0

IV , 2) (-, +, +, -)

V I , 2) (-, -, +, +) 2

Table 6. The correspondence with n i = 5

I , 1) (+, +, +, +, -) 3 (T

IV , 2) (+, -, +, +, -) 1 (T

V , 1) (-, +, +, +, +) 4 (T

V , 2) (-, +, +, +, -)

V II , 2) (+, -, -,

X , 2) (-, -, +, +, +) 3

T l1,• T l2,• there exists a subundle O(1) ⊂ E as below:

there exists a subundle O ⊂ E as below:

[