Fundamental group of Galois covers of degree 6 surfaces
Meirav Amram, Cheng Gong, Uriel Sinichkin, Sheng-Li Tan, Wan-Yuan Xu, Michael Yoshpe
aa r X i v : . [ m a t h . AG ] J a n Fundamental group of Galois covers of degree 6 surfaces
Meirav Amram , Cheng Gong , Uriel Sinichkin , Sheng-Li Tan , Wan-Yuan Xu , and MichaelYoshpe
61, 6
Department of Mathematics, Shamoon College of Engineering, Ashdod, Israel Department of Mathematics, Soochow University, Suzhou 215006, Jiangsu, P. R. China School of Mathematical Sciences, Tel Aviv University, Tel Aviv, Israel School of Mathematical Sciences, Shanghai Key Labaratory of PMMP, East China Normal University,Shanghai 200241, P. R. China Department of Mathematics, Shanghai Normal University, Shanghai 200234, P. R. China
Abstract
In this paper we consider the Galois covers of algebraic surfaces of degree 6, with all associatedplanar degenerations. We compute the fundamental groups of those Galois covers, using their de-generation. We show that for 8 types of degenerations the fundamental group of the Galois cover isnon-trivial and for 20 types it is trivial. Moreover, we compute the Chern numbers of all the surfaceswith this type of degeneration and prove that the signatures of all their Galois covers are negative.We formulate a conjecture regarding the structure of the fundamental groups of the Galois coversbased on our findings.
Email addresses: Meirav Amram (corresponding author): [email protected]; Cheng Gong: [email protected];Uriel Sinichkin: [email protected]; Wan-Yuan Xu: [email protected] Tan: [email protected]; Michael Yoshpe: [email protected] Mathematics Subject Classification. 14D05, 14D06, 14H30, 14J10, 20F36.
Key words : Degeneration, generic projection, Galois cover, braid monodromy, fundamental group ontents X degenerates to U , , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.4 X degenerates to U , , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.5 X degenerates to U ∪ , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 References 23A Classification of the cases 25B Full calculations 29
B.1 X degenerates to U , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29B.2 X degenerates to U , , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31B.3 X degenerates to U , , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33B.4 X degenerates to U , , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35B.5 X degenerates to U , , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35B.6 X degenerates to U , , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37B.7 X degenerates to U , , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39B.8 X degenerates to U , , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41B.9 X degenerates to U , , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43B.10 X degenerates to U , , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45B.11 X degenerates to U , , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46B.12 X degenerates to U , , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48B.13 X degenerates to U , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50B.14 X degenerates to U , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53B.15 X degenerates to U , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55B.16 X degenerates to U , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58B.17 X degenerates to U , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 .18 X degenerates to U , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63B.19 X degenerates to U , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65B.20 X degenerates to U ∪ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68B.21 X degenerates to U , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71B.22 X degenerates to U , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73B.23 X degenerates to U , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75B.24 X degenerates to U ∪ , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78B.25 X degenerates to U ∪ , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78B.26 X degenerates to U ∪ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81B.27 X degenerates to U . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84B.28 X degenerates to U ∪ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86B.29 X degenerates to U . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 Introduction
Classification of algebraic surfaces has been one of the most significant problems guiding the devel-opment of algebraic geometry throughout its history. In modern mathematics this classification hasoften been studied through its moduli space (see, for example, the works of Catanese [13, 14]).Recent progress in this area is due to the study of invariants related to the Galois covers of suchsurfaces. To an algebraic surface X of degree n embedded in a projective space CP N , we attach aGalois cover. The Galois cover X Gal is the Zariski closure of the fibered product, with respect toa generic projection to CP , of n copies of X , where the generalized diagonal is excluded. For adetailed definition of the Galois cover, see Section 2.Galois covers and their related fundamental groups were studied by Gieseker [17], Liedtke [19],Moishezon-Teicher [21], A.-Goldberg [3], A.-Teicher-Vishne [10]. Works on surfaces of degrees 4and 5 and their Galois covers were done by A.-Lehman-Shwartz-Teicher [7] and A.-G.-Teicher-X.[6], respectively. Lately, the fundamental group of Galois covers were computed for surfaces withZappatic singularity by A.-G.-T.-Teicher-X. in [5] and for embeddings of CP × T of higher degreesby A.-T.-X.-Y. in [9]. In all those works, the Galois covers provide a way to construct importantinvariants of the base surfaces and contribute to their classification. They are also primary toolused to construct an example of a simply-connected surface of positive index [21], thus disprovingBogomolov’s watershed conjecture.In this paper we study 29 surfaces of degree 6 and their Galois covers. It is a continuation ofthe work on degenerations of degree 5 [6]. We study braid monodromy, the fundamental groups ofthe complements of the branch curves and of the Galois covers of degree 6 surfaces with nice planardegenerations. Degenerations were studied by Calabri-Ciliberto-Flamini-Miranda in [12], Ciliberto-Lopez-Miranda in [15], A.-G.-T.-Teicher-X. in [5], and in other papers such as [1, 2, 3]. The braidmonodromy algorithm was presented by Moishezon-Teicher in [22, 23]. Fundamental groups of thecomplements of branch curves were studied for their own sake by Zariski in [25] and by Auroux-Donaldson-Katzarkov-Yotov in [11].Below we describe briefly the process taken in this paper; for full details see Section 2. Thecomputation of the Galois cover and its invariants begins with the inspection of the branch curve S .In general, it is difficult to describe S for a given projection X onto CP . In order to overcome thisobstacle, we degenerate X to a union of planes X , which then degenerates the branch curve intoan arrangement of lines S . Once we have this line arrangement, we can use the reverse process ofregeneration, which is described by the so called ”regeneration Lemmas” from [22], to recover partialinformation regarding the branch curve S of X .This suffices to compute the braid monodromy of S by the Moishezon-Teicher algorithm. Havingthe braid monodromy of S , one can apply the van-Kampen Theorem [18] to calculate the fundamentalgroup π ( CP − S ) of the complement of S in CP . Then it is possible to calculate the fundamental roup π ( X Gal ) of the Galois cover X Gal of X by considering an exact sequence (see [21]),0 → π ( X Gal ) → e G → S n → , (1)where e G := π ( CP − S ) / h Γ i i . The group π ( X Gal ) is the kernel of the natural projection e G → S n .The above process computes the correct fundamental group π ( X Gal ), because Moishezon-Teichershowed in [21] that when the complex structure of X changes continuously, π ( X Gal ) does not change.Other important invariants of surfaces are their Chern numbers c and c (see, for example,Manetti’s work [20]). Both c ( X Gal ) and c ( X Gal ) can be computed from the braid monodromy (see[21]).We have considered surfaces that have degeneration with planar representation (see Definition3.1), of which there are 29 types (see [4, Appendix A]). The main result of this paper is that π ( X Gal )is not trivial in eight out of those 29 types, trivial in 20 other cases, and for one case the questionwhether π ( X Gal ) is trivial or not remains open (see Table 1 for details). As an application, wecompute the Chern numbers for all 29 Galois covers. We find degenerations of surfaces that havethe same Chern numbers, i.e., the related surfaces are in the same component of the moduli spaces.Moreover, the signatures of all 29 types, that are computed using the Chern number of those Galoiscovers, are all negative.This paper is organized as follows: In Section 2, we explain important methods and present thefundamental and necessary background that we use in this paper. In Section 3 we state and proveauxiliary lemmas we used in the calculations and provide the full computation of three of the caseswe compute, to show an example of our methods. In Section 4 we present the results of all ourcalculations, including the computation of the Chern numbers and signatures of the Galois covers ofthe surfaces we consider. We include an appendix with a proof of the classification of the 29 casesof surfaces degenerating to degree 6 objects. Moreover, we give full calculations of the fundamentalgroups of the Galois covers.
Acknowledgements:
The authors are grateful to Guo Zhiming for useful discussions about hiswork on the classification of the degree six surfaces. This research was supported by the ISF-NSFCjoint research program (Grant No. 2452/17) .
In this section we give the background needed for the computations that appear in later sections.Additionally, we introduce all the notations used in the paper.Let X be a projective algebraic surface embedded in projective space CP N , for some N . Considera generic projection f : CP N → CP . The restriction of f | X is branched along a curve S ⊆ CP . he branch curve S can tell a lot about X , but it is difficult to describe it explicitly. To tackle thisproblem we consider degeneration of X , defined as follows. Definition 2.1.
Let ∆ be the unit disc, and let
X, Y be projective algebraic surfaces. Let p : Y → CP and p ′ : X → CP be projective embeddings. We say that p ′ is a projective degeneration of p ifthere exists a flat family π : V → ∆ and an embedding F : V → ∆ × CP , such that F composedwith the first projection is π , and:(a) π − (0) ≃ X ;(b) there is a t = 0 in ∆ such that π − ( t ) ≃ Y ;(c) the family V − π − (0) → ∆ − π − (0), F = 0 × p ′ under the identification of π − (0) with X ;(e) restricting to π − ( t ), F = t × p under the identification of π − ( t ) with Y .We construct a degeneration of X into a union of planes, as a sequence of partial degenerations X := X r ❀ X r − ❀ · · · X r − i ❀ X r − ( i +1) ❀ · · · ❀ X . Consider generic projections π ( i ) : X i → CP with the branch curve S i for 0 ≤ i ≤ r . Note that S i − is a degeneration of S i .Because X is a union of planes, its projection S is a line arrangement. Locally around eachsingular point, S is defined by the multiplicity of this singularity. We call a singular point ofmultiplicity k , a k -point . A 1-point always comes from the intersection of 2 planes in X . Similarly,a 2-point is always the projection of the intersection of 3 planes P , P , P in X , with P and P intersecting only in the singular point (and not in a line).Next, for a 3-point in S , two options can occur. Either the 3-point is an intersection of threeplanes in X , with every pair of planes intersecting in a line, or it is the intersection of four planes P , . . . , P in X , s.t. P i and P j intersect in a line only if | i − j | = 1. We call the first option an inner -point and the second an outer -point . For singularities of higher multiplicities additional types ofsingularities can occur in S , but in the arrangements we consider in the current work, only 2 typescan happen for each multiplicity. Definition 2.2.
We call a k -point that is the intersection of k + 1 planes P , . . . , P k +1 , s.t. P i intersect P j in a line if and only if | i − j | = 1, an outer k -point .We call a k -point that is the intersection of k planes P , . . . , P k , s.t. P i intersects P j in a line if | i − j | = 1 and additionally P intersects P k , an inner k -point .Note that those singularities were considered in [12], where the inner k -point is denoted by E k and the outer k -point is denoted by R k . Example 2.1.
We explain the above notions on the degeneration using Figure 1. The figure is aschematic representation of X . The components of X are represented by triangles and their lines(common edges). So P and P intersect in a line indexed 1, P and P intersect in a line indexed 2, P and P are disjoint, and so on. he vertices of the diagram that have inner edges connected to them are the singular points of X . Vertex 6 is a 1-point, vertices 2 and 3 are 2-points, vertex 4 is an inner 3-point and vertex 7 isan outer 4-point.
12 345 674 3 25 61 P P P P P P Figure 1:
An example of degeneration into a union of planes.
All the degenerations considered in the current work appear in Figure 2.We note that 1- and 2-points were considered in [1, 8, 21], 3-points were considered in [1], 4-pointswere considered in [1, 8], and 5-points were considered in [16]. The regeneration process shown belowcan be quite difficult for a large k , but work has been done for some specific values (see [16, 5, 2] for5-, 6-, and 8-points, respectively).One of the principal tools we use is a reverse process of degeneration, called regeneration . Usingthis tool, which was described in [22] as regeneration Lemmas, we can recover S i from S i − . Applyingit multiple times we can recover the original branch curve S from the line arrangement S . In thefollowing diagram, we illustrate this process. X ⊆ CP N degeneration −−−−−−−−→ X ⊆ CP M generic projection y y generic projection S ⊆ CP ←−−−−−−−− regeneration S ⊆ CP A line in S regenerates either to a conic or a double line. The resulting components of thepartial regeneration are tangent to each other. To get a transversal intersection of components, weregenerate further, and this gives us three cusps for each tangency point (see [22, 23] for more details).Therefore, the regenerated branch curve S is a cuspidal curve.Using the notations of [23] as well as the additional notation Z i i ′ ,j j ′ introduced here, we candenote the braids related to S as follows:(1) for a branch point, Z j j ′ is a counterclockwise half-twist of j and j ′ along a path below the realaxis,
2) for nodes, Z i,j j ′ = Z i j · Z i j ′ and Z i i ′ ,j j ′ = Z i ′ j ′ · Z i j ′ · Z i ′ j · Z i j ,(3) for cusps, Z i,j j ′ = Z i j · ( Z i j ) Z j j ′ · ( Z i j ) Z − j j ′ .A singular point of S gives rise to braids (1) − (3) only locally. To get the final braids, one needsto perform a conjugation, which we denote as a b = b − ab . In several places we use the notation ¯ a where a is a braid that means the same braid as a but above the real axis.Denote G := π ( CP − S ) and its standard generators as Γ , Γ ′ , . . . , Γ m , Γ ′ m . By the van KampenTheorem [18] we can get a presentation of G by means of generators { Γ j , Γ ′ j } and relations of thetypes:(1) for a branch point, Z j j ′ corresponds to the relation Γ j = Γ ′ j ,(2) for nodes, Z i j corresponds to [Γ i , Γ j ] = Γ i Γ j Γ − i Γ − j = e ,(3) for cusps, Z i j corresponds to h Γ i , Γ j i = Γ i Γ j Γ i Γ − j Γ − i Γ − j = e .To get all the relations, we write the braids in a product and collect all the relations that correspondto the different factors. See [22, 23] for full treatment of the subject.To each list of relations we add the projective relation Q j = m Γ ′ j Γ j = e . Moreover, in some casesin the paper, we have parasitic intersections that induce commutative relations. These intersectionscome from lines in X that do not intersect, but when projecting X onto CP , they will intersect(see details in [21]).Our techniques also allow us to compute fundamental groups of Galois covers. We recall from[21] that if f : X → CP is a generic projection of degree n , then X Gal , the Galois cover, is definedas follows: X Gal = ( X × CP . . . × CP X ) − △ , where the product is taken n times, and △ is the diagonal. To apply a theorem of Moishezon-Teicher[21], we define e G := G/ h Γ i i . Then, there is an exact sequence0 → π ( X Gal ) → e G → S n → , (2)where the second map takes the generators Γ i of G to transpositions in the symmetric group S n according to the order of the lines in the degenerated surface. We thus obtain a presentation ofthe fundamental group of the Galois cover. We next simplify the relations to produce a canonicalpresentation that identifies with π ( X Gal ), using the theory of Coxeter covers of the symmetric groups.For more details see the proof of Theorem 3.6.
In this section, we define the degenerations we are studying in this paper. The total number of suchdegenerations of degree 6 is 29, as shown in [4, Appendix A], and as depicted in Figure 2. We nextcompute the fundamental groups of their Galois covers. he full calculations for all the cases are quite long and would not fit in the current paper. Wepresent here three examples of possible choices for X : U , , , U , , and U ∪ , . Those examplesindicate the techniques we are using. The computations of π ( X Gal ) for the 26 additional cases, aswell as the computation of the Chern numbers of all 29 cases, are presented in [4, Appendix B].
Here we will explain the types of surface degenerations we have considered and the way to enumeratethem.
Definition 3.1.
A degeneration of smooth toric surface X into a union of planes X is said to havea planar representation if:(1) No three planes in X intersect in a line.(2) There exists a simplicial complex with connected interior, embedded in R , s.t. its two dimen-sional cells correspond bijectively to irreducible components of X and this bijection preservesan incidence relation.This choice is not the most general that can be made, but most degenerations of surfaces thatare classically of interest are of this form. It is also the choice made in [6, 7].Note that degenerations that have planar representation of bounded degree can be enumeratedrecursively, because the planar simplicial complexes can be enumerated. Such enumeration for degree6 is presented in [4, Appendix A].The list of degenerations we get from this enumeration, and thus the degenerations we are con-sidering in the paper, are presented in Figure 2. The labeling of the possible choices for X came from the combinatorial classification (see [4, Appendix A]). , U , , U , , U , , U , , U , , U , , U , , U , , U , , U , , U , , U , U , U , U , U , U , U , U ∪ U , U , U , U ∪ , U ∪ , U ∪ U U ∪ U Figure 2:
The collection of all the degenerations we inspect in the current work. .2 General setup and useful lemmas In this subsection we prove couple of results we will use later in the computation.We will work with a dual graph T of X , which is defined when vertices of T are in bijection withthe planes in X , and the vertices corresponding to the planes P and P are connected by an edgeif P and P intersect in a line.First, we will provide the principal tool we will use to prove that π ( X Gal ) is not trivial in certaincases.
Lemma 3.2.
Let ˆ G := e G/ h Γ i = Γ ′ i i . If T has a vertex of valency of at least 3 that is not part of acycle in T , then the kernel of the natural projection ˆ G → S is not trivial. In particular, π ( X Gal ) isnot trivial in this case.Proof. Let i, j, k be three distinct edges connected to the vertex p of T that satisfy the propertiesrequired in the Lemma. Because X has a planar representation (Definition 3.1), T is a planar graph.Thus, the three lines in X corresponding to i, j, k cannot meet in a point. By [24], the kernel of thenatural homomorphism ˆ G → S contains the relation [Γ i , Γ j Γ k Γ j ] = e , so we are left to show thatthis relation is not trivial in ˆ G .To that end, we take the quotient of ˆ G by the group normally generated by { Γ l | l / ∈ { i, j, k }} . Wenow can inspect what can happen to the relations that arise from branch points, nodes, and cusps.Note that in every such relation, at most two instances of Γ i , Γ j , Γ k can appear, because the lines i, j, k in X do not pass through one point. Branch points:
A branch point relation with only two of the generators Γ i , Γ j , Γ k is either re-dundant or equivalent to a commutation relation between two of the generators. Because notwo such generators can commute (otherwise it would contradict the existence of the naturalhomomorphism e G → S , all those relations are trivial in ˆ G . Nodes:
This case is identical to the previous - the only commutation relations with only two gen-erators are redundant or a commutation between two generators that do not hold in ˆ G . Cusps:
Cusps give rise to triple relations h w , w i for some w , w ∈ ˆ G , which obviously cannot leadto the fork relation [Γ i , Γ j Γ k Γ j ] = e .We next list some results that will allow us to show Γ i = Γ ′ i in e G . This will be the principal toolat our disposal to show that the group π ( X Gal ) is trivial.
Lemma 3.3.
Let p be a -point in X with lines i and j . Then Γ i = Γ ′ i in e G iff Γ j = Γ ′ j in e G .Proof. Vertex p is a 2-point. The braid monodromy corresponding to this point is: f ∆ p = ( Z j j ′ ) Z i i ′ ,j · Z i i ′ ,j . ∆ p gives rise to the following relations: h Γ i , Γ j i = h Γ ′ i , Γ j i = h Γ − i Γ ′ i Γ i , Γ j i = e (3)Γ ′ j = Γ j Γ ′ i Γ i Γ j Γ − i Γ ′− i Γ − j . (4)Now, if Γ i = Γ ′ i , relation (4) becomes Γ j = Γ ′ j . On the other hand, if Γ j = Γ ′ j , we can deducefrom (3) and (4) that Γ j Γ ′ i Γ j = Γ ′ i Γ j Γ ′ i = Γ i Γ j Γ i = Γ j Γ i Γ j and so we get Γ i = Γ ′ i , as needed. Lemma 3.4.
Let p be an inner -point (see Definition 2.2) in X with lines i < j < k (see Figure3). If either Γ j = Γ ′ j or Γ k = Γ ′ k in e G then Γ l = Γ ′ l for all l ∈ { i, j, k } .Moreover, Γ i = Γ ′ i is always present in this case. p ij k Figure 3:
The vertex p in Lemma 3.4.Proof. Vertex p is an inner 3-point. Its braid monodromy is f ∆ p = Z i ′ ,j j ′ · Z i,j j ′ · ( Z j k ′ ) Z j j ′ Z i ′ ,j j ′ · ( Z j ′ k ) Z j j ′ Z k k ′ Z i ′ ,j j ′ · ( Z j k ′ ) Z j j ′ Z i,j j ′ · ( Z j ′ k ) Z j j ′ Z k k ′ Z i,j j ′ . These braids give rise to the following relations in G : h Γ ′ i , Γ j i = h Γ ′ i , Γ ′ j i = h Γ ′ i , Γ − j Γ ′ j Γ j i = e (5) h Γ i , Γ j i = h Γ i , Γ ′ j i = h Γ i , Γ − j Γ ′ j Γ j i = e (6)Γ ′ j Γ j Γ ′ i Γ j Γ ′− i Γ − j Γ ′ j − = Γ ′ k (7)Γ ′ j Γ j Γ ′ i Γ ′ j Γ ′− i Γ − j Γ ′ j − = Γ ′ k Γ k Γ ′ k − (8)Γ ′ j Γ j Γ i Γ j Γ − i Γ − j Γ ′ j − = Γ ′ k (9)Γ ′ j Γ j Γ i Γ ′ j Γ − i Γ − j Γ ′ j − = Γ ′ k Γ k Γ ′− k . (10)First, we will use (7) and (8) to get that Γ j = Γ ′ j if and only if Γ k = Γ ′ k . Indeed, if Γ k = Γ ′ k thenwe get that the right sides of those equations are equal, so by equating the left sides, we get Γ j = Γ ′ j .Conversely, if Γ j = Γ ′ j , we get that the left sides of (7) and (8) are equal, so by equating the rightsides we get that Γ k = Γ ′ k . or the second part of the statement, we equate the left sides of (7) and (9) and then use (5) and(6), similarly to the proof of Lemma 3.3. Lemma 3.5.
Let p be an outer -point in X with lines i < j < k (see Figure 4). If either Γ i = Γ ′ i or Γ j = Γ ′ j in e G then Γ l = Γ ′ l for all l ∈ { i, j, k } . p ijk Figure 4:
The vertex p in Lemma 3.5.Proof. Vertex p is an outer 3-point. The braid monodromy corresponding to this 3-point is: f ∆ p = Z i i ′ ,k k ′ · Z i ′ ,j j ′ · ( Z i i ′ ) Z i ′ ,j j ′ · ( Z j j ′ ,k ) Z i ′ ,j j ′ · ( Z k k ′ ) Z j j ′ ,k Z i ′ ,j j ′ . f ∆ p thus gives rise to the following relations: h Γ ′ i , Γ j i = h Γ ′ i , Γ ′ j i = h Γ ′ i , Γ − j Γ ′ j Γ j i = e (11)Γ i = Γ ′ j Γ j Γ ′ i Γ − j Γ ′ j − (12) h Γ k , Γ ′ j Γ j Γ ′ i Γ j Γ ′ i − Γ − j Γ ′ j − i = h Γ k , Γ ′ j Γ j Γ ′ i Γ ′ j Γ ′ i − Γ − j Γ ′ j − i == h Γ k , Γ ′ j Γ j Γ ′ i Γ − j Γ ′ j Γ j Γ ′ i − Γ − j Γ ′ j − i = e (13)Γ ′ k = Γ k Γ ′ j Γ j Γ ′ i Γ ′ j Γ j Γ ′ i − Γ − j Γ ′ j − Γ k Γ ′ j Γ j Γ ′ i Γ − j Γ ′ j − Γ ′ i − Γ − j Γ ′ j − Γ − k (14)[Γ i , Γ k ] = [Γ i , Γ ′ k ] = [Γ ′ i , Γ k ] = [Γ ′ i , Γ ′ k ] = e. (15)If Γ j = Γ ′ j we get Γ i = Γ ′ i from (12) and then Γ k = Γ ′ k from (14). If Γ i = Γ ′ i , we get Γ j = Γ ′ j similarly to the proof of Lemma 3.3, and then Γ k = Γ ′ k from (14). .3 X degenerates to U , ,
12 3 45 6 78 123 4 5
Figure 5:
The arrangement of planes U , , . Theorem 3.6. If X degenerates to U , , , then π ( X Gal ) is not trivial.Proof. The branch curve S in CP is an arrangement of five lines. We regenerate each vertex inturn and compute the group G .Vertex 5 is a 1-point that gives rise to braid Z ′ and to the following relation in G :Γ = Γ ′ . (16)Vertex 7 is a 1-point that gives rise to braid Z ′ and to the following relation in G :Γ = Γ ′ . (17)Vertex 2 is a 2-point. The braid monodromy corresponding to this point is: f ∆ = ( Z ′ ) Z
22 2 ′ , · Z
32 2 ′ , . f ∆ gives rise to the following relations: h Γ , Γ i = h Γ ′ , Γ i = h Γ − Γ ′ Γ , Γ i = e (18)Γ ′ = Γ Γ ′ Γ Γ Γ − Γ ′− Γ − . (19)Vertex 3 is an outer 3-point. The braid monodromy corresponding to this 3-point is: f ∆ = Z
23 3 ′ , ′ · Z ′ , ′ · ( Z ′ ) Z ′ , ′ · ( Z
34 4 ′ , ) Z ′ , ′ · ( Z ′ ) Z
24 4 ′ , Z ′ , ′ . f ∆ thus gives rise to the following relations: h Γ ′ , Γ i = h Γ ′ , Γ ′ i = h Γ ′ , Γ − Γ ′ Γ i = e (20) = Γ ′ Γ Γ ′ Γ − Γ ′ − (21) h Γ , Γ ′ Γ Γ ′ Γ Γ ′ − Γ − Γ ′ − i = h Γ , Γ ′ Γ Γ ′ Γ ′ Γ ′ − Γ − Γ ′ − i == h Γ , Γ ′ Γ Γ ′ Γ − Γ ′ Γ Γ ′ − Γ − Γ ′ − i = e (22)Γ ′ = Γ Γ ′ Γ Γ ′ Γ ′ Γ Γ ′ − Γ − Γ ′ − Γ Γ ′ Γ Γ ′ Γ − Γ ′ − Γ ′ − Γ − Γ ′ − Γ − (23)[Γ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ] = [Γ ′ , Γ ′ ] = e. (24)Vertex 6 is an outer 3-point. The braid monodromy corresponding to this 3-point is: f ∆ = Z
21 1 ′ , ′ · Z ′ , ′ · ( Z ′ ) Z ′ , ′ · ( Z
32 2 ′ , ) Z ′ , ′ · ( Z ′ ) Z
22 2 ′ , Z ′ , ′ . f ∆ thus gives rise to the following relations: h Γ ′ , Γ i = h Γ ′ , Γ ′ i = h Γ ′ , Γ − Γ ′ Γ i = e (25)Γ = Γ ′ Γ Γ ′ Γ − Γ ′ − (26) h Γ , Γ ′ Γ Γ ′ Γ Γ ′ − Γ − Γ ′ − i = h Γ , Γ ′ Γ Γ ′ Γ ′ Γ ′ − Γ − Γ ′ − i == h Γ , Γ ′ Γ Γ ′ Γ − Γ ′ Γ Γ ′ − Γ − Γ ′ − i = e (27)Γ ′ = Γ Γ ′ Γ Γ ′ Γ ′ Γ Γ ′ − Γ − Γ ′ − Γ Γ ′ Γ Γ ′ Γ − Γ ′ − Γ ′ − Γ − Γ ′ − Γ − (28)[Γ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ] = [Γ ′ , Γ ′ ] = e. (29)We also have the following parasitic and projective relations:[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (30)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (31)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (32)Γ ′ Γ Γ ′ Γ Γ ′ Γ Γ ′ Γ Γ ′ Γ = e. (33)By Lemma 3.5 we get Γ = Γ ′ and Γ = Γ ′ . Then by Lemma 3.3, we get Γ = Γ ′ .Thus e G is generated by { Γ i | i = 1 , . . . , } modulo the following relations:Γ = Γ = Γ = Γ = Γ = e (34) h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = e (35)[Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = e. (36)By Lemma 3.2, π ( X Gal ) is not trivial. .4 X degenerates to U , , Figure 6:
The arrangement of planes U , , . Theorem 3.7. If X degenerates to U , , , then π ( X Gal ) is trivial.Proof. The branch curve S in CP is an arrangement of five lines. We regenerate each vertex inturn and compute the group G .Vertex 1 is a 1-point that gives rise to braid Z ′ and to the following relation in G :Γ = Γ ′ . (37)Vertex 3 is a 1-point that gives rise to braid Z ′ and to the following relation in G :Γ = Γ ′ . (38)Vertex 5 is a 1-point that gives rise to braid Z ′ and to the following relation in G :Γ = Γ ′ . (39)Vertex 6 is a 1-point that gives rise to braid Z ′ and to the following relation in G :Γ = Γ ′ . (40)Vertex 7 is a 2-point. The braid monodromy corresponding to this point is: f ∆ = ( Z ′ ) Z
24 4 ′ , · Z
34 4 ′ , . f ∆ gives rise to the following relations: h Γ , Γ i = h Γ ′ , Γ i = h Γ − Γ ′ Γ , Γ i = e (41)Γ ′ = Γ Γ ′ Γ Γ Γ − Γ ′− Γ − . (42)Vertex 4 is a 4-point. Its braid monodromy is e ∆ = Z
33 3 ′ , · ( Z
21 1 ′ , ) Z −
22 2 ′ , · Z
22 2 ′ , · ¯ Z
21 1 ′ , ′ · ¯ Z
22 2 ′ , ′ · ( Z ′ ) Z
23 3 ′ , · Z ′ , ′ · ( Z ′ ) Z ′ , ′ · ( Z
32 2 ′ , ) Z ′ , ′ Z
23 3 ′ , · ( Z ′ ) Z
22 2 ′ , Z ′ , ′ Z
23 3 ′ , · ( Z
21 1 ′ , ′ ) Z
23 3 ′ , . t gives rise to the following relations: h Γ , Γ i = h Γ ′ , Γ i = h Γ − Γ ′ Γ , Γ i = e (43)[Γ ′ Γ Γ Γ − Γ ′− , Γ ] = [Γ ′ Γ Γ ′ Γ − Γ ′− , Γ ] = e (44)[Γ , Γ ] = [Γ ′ , Γ ] = e (45)[Γ ′ Γ Γ ′ Γ Γ Γ − Γ ′− Γ − Γ ′− , Γ − Γ ′ Γ ] = [Γ ′ Γ Γ ′ Γ Γ ′ Γ − Γ ′− Γ − Γ ′− , Γ − Γ ′ Γ ] = e (46)[Γ ′ Γ Γ Γ − Γ ′− , Γ − Γ ′ Γ ] = [Γ ′ Γ Γ ′ Γ − Γ ′− , Γ − Γ ′ Γ ] = e (47)Γ Γ ′ Γ Γ Γ − Γ ′− Γ − = Γ ′ (48) h Γ ′ , Γ i = h Γ ′ , Γ ′ i = h Γ ′ , Γ − Γ ′ Γ i = e (49)Γ = Γ ′ Γ Γ ′ Γ − Γ ′− (50) h Γ ′ Γ Γ ′ Γ Γ ′− Γ − Γ ′− , Γ Γ Γ − i = h Γ ′ Γ Γ ′ Γ ′ Γ ′− Γ − Γ ′− , Γ Γ Γ − i == h Γ ′ Γ Γ ′ Γ − Γ ′ Γ Γ ′− Γ − Γ ′− , Γ Γ Γ − i = e (51)Γ ′ Γ Γ ′ Γ − Γ ′− Γ ′− Γ − Γ ′− Γ Γ − Γ ′ Γ Γ − Γ ′ Γ Γ ′ Γ ′ Γ Γ ′− Γ − Γ ′− = Γ Γ Γ − (52)[Γ , Γ Γ Γ − ] = [Γ ′ , Γ Γ Γ − ] = [Γ , Γ Γ ′ Γ − ] = [Γ ′ , Γ Γ ′ Γ − ] = e. (53)We also have the following parasitic and projective relations:[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (54)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (55)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (56)Γ ′ Γ Γ ′ Γ Γ ′ Γ Γ ′ Γ Γ ′ Γ = e. (57)By Lemma 3.3, we get Γ = Γ ′ . e G is thus generated by { Γ i | i = 1 , . . . , } with the following relations:Γ = Γ = Γ = Γ = Γ = e (58) h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = e, (59)[Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = e. (60)Thus, e G ∼ = S , therefore π ( X Gal ) is trivial. .5 X degenerates to U ∪ , Figure 7:
The arrangement of planes U ∪ , . Theorem 3.8. If X degenerates to U ∪ , , then π ( X Gal ) is trivial.Proof. The branch curve S in CP is an arrangement of seven lines. We regenerate each vertex inturn and compute the group G .Vertex 6 is a 1-point that gives rise to braid Z ′ and to the following relation in G :Γ = Γ ′ . (61)Vertex 2 is an inner 3-point. Its braid monodromy is f ∆ = Z ′ , ′ · Z , ′ · ( Z ′ ) Z
25 5 ′ Z ′ , ′ · ( Z ′ ) Z
25 5 ′ Z
26 6 ′ Z ′ , ′ · ( Z ′ ) Z
25 5 ′ Z , ′ · ( Z ′ ) Z
25 5 ′ Z
26 6 ′ Z , ′ . These braids give rise to the following relations in G : h Γ ′ , Γ i = h Γ ′ , Γ ′ i = h Γ ′ , Γ − Γ ′ Γ i = e (62) h Γ , Γ i = h Γ , Γ ′ i = h Γ , Γ − Γ ′ Γ i = e (63)Γ ′ Γ Γ ′ Γ Γ ′− Γ − Γ ′ − = Γ ′ (64)Γ ′ Γ Γ ′ Γ ′ Γ ′− Γ − Γ ′ − = Γ ′ Γ Γ ′ − (65)Γ ′ Γ Γ Γ Γ − Γ − Γ ′ − = Γ ′ (66)Γ ′ Γ Γ Γ ′ Γ − Γ − Γ ′ − = Γ ′ Γ Γ ′− . (67)Vertex 3 is an outer 3-point. The braid monodromy corresponding to this 3-point is: f ∆ = Z
21 1 ′ , ′ · Z ′ , ′ · ( Z ′ ) Z ′ , ′ · ( Z
36 6 ′ , ) Z ′ , ′ · ( Z ′ ) Z
26 6 ′ , Z ′ , ′ . ∆ thus gives rise to the following relations: h Γ ′ , Γ i = h Γ ′ , Γ ′ i = h Γ ′ , Γ − Γ ′ Γ i = e (68)Γ = Γ ′ Γ Γ ′ Γ − Γ ′ − (69) h Γ , Γ ′ Γ Γ ′ Γ Γ ′ − Γ − Γ ′ − i = h Γ , Γ ′ Γ Γ ′ Γ ′ Γ ′ − Γ − Γ ′ − i == h Γ , Γ ′ Γ Γ ′ Γ − Γ ′ Γ Γ ′ − Γ − Γ ′ − i = e (70)Γ ′ = Γ Γ ′ Γ Γ ′ Γ ′ Γ Γ ′ − Γ − Γ ′ − Γ Γ ′ Γ Γ ′ Γ − Γ ′ − Γ ′ − Γ − Γ ′ − Γ − (71)[Γ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ] = [Γ ′ , Γ ′ ] = e. (72)Vertex 4 is an outer 3-point. The braid monodromy corresponding to this 3-point is: f ∆ = Z
21 1 ′ , ′ · Z ′ , ′ · ( Z ′ ) Z ′ , ′ · ( Z
32 2 ′ , ) Z ′ , ′ · ( Z ′ ) Z
22 2 ′ , Z ′ , ′ . f ∆ thus gives rise to the following relations: h Γ ′ , Γ i = h Γ ′ , Γ ′ i = h Γ ′ , Γ − Γ ′ Γ i = e (73)Γ = Γ ′ Γ Γ ′ Γ − Γ ′ − (74) h Γ , Γ ′ Γ Γ ′ Γ Γ ′ − Γ − Γ ′ − i = h Γ , Γ ′ Γ Γ ′ Γ ′ Γ ′ − Γ − Γ ′ − i == h Γ , Γ ′ Γ Γ ′ Γ − Γ ′ Γ Γ ′ − Γ − Γ ′ − i = e (75)Γ ′ = Γ Γ ′ Γ Γ ′ Γ ′ Γ Γ ′ − Γ − Γ ′ − Γ Γ ′ Γ Γ ′ Γ − Γ ′ − Γ ′ − Γ − Γ ′ − Γ − (76)[Γ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ] = [Γ ′ , Γ ′ ] = e. (77)Vertex 1 is an inner 4-point. Its braid monodromy appears in [8]. The corresponding relations in G are: h Γ ′ , Γ i = h Γ ′ , Γ ′ i = h Γ ′ , Γ − Γ ′ Γ i = e (78) h Γ , Γ i = h Γ ′ , Γ i = h Γ − Γ ′ Γ , Γ i = e (79)[Γ ′ Γ Γ ′ Γ − Γ ′ − , Γ ] = e (80)[Γ ′ Γ Γ ′ Γ − Γ ′ − , Γ − Γ ′ − Γ − Γ ′ Γ Γ ′ Γ ] = e (81) h Γ , Γ i = h Γ , Γ ′ i = h Γ , Γ − Γ ′ Γ i = e (82) h Γ , Γ − Γ ′ Γ i = h Γ ′ , Γ − Γ ′ Γ i = h Γ − Γ ′ Γ , Γ − Γ ′ Γ i = e (83)[Γ ′ Γ Γ Γ − Γ ′ − , Γ − Γ ′ Γ ] = e (84)[Γ ′ Γ Γ Γ − Γ ′ − , Γ − Γ ′ − Γ − Γ ′ − Γ Γ ′ Γ Γ ′ Γ ] = e (85)Γ ′ Γ Γ ′ Γ Γ ′− Γ − Γ ′ − = Γ Γ ′ Γ − (86)Γ ′ Γ Γ ′ Γ ′ Γ ′− Γ − Γ ′ − = Γ Γ ′ Γ Γ ′ − Γ − (87) ′ Γ Γ Γ Γ − Γ − Γ ′ − = Γ − Γ ′ Γ Γ ′ Γ − Γ ′ − Γ (88)Γ ′ Γ Γ Γ ′ Γ − Γ − Γ ′ − = Γ − Γ ′ Γ Γ ′ Γ Γ ′− Γ − Γ ′ − Γ . (89)We also have the following parasitic and projective relations:[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (90)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (91)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (92)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (93)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (94)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (95)Γ ′ Γ Γ ′ Γ Γ ′ Γ Γ ′ Γ Γ ′ Γ Γ ′ Γ Γ ′ Γ = e. (96)Substituting (61) and equating (86) and (87), we get Γ = Γ ′ . By Lemma 3.4, we have Γ = Γ ′ and Γ = Γ ′ . Then by Lemma 3.5, we get Γ = Γ ′ , Γ = Γ ′ and Γ = Γ ′ . e G is thus generated by { Γ i | i = 1 , . . . , } with the following relations:Γ = Γ = Γ = Γ = Γ = Γ = Γ = e (97) h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = (98) h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = e [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] == [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = e (99)Γ = Γ Γ Γ (100)Γ Γ Γ = Γ Γ Γ . (101)By Remark 2.8 in [24], we get also the relations[Γ , Γ Γ Γ ] = [Γ , Γ Γ Γ ] = [Γ , Γ Γ Γ ] = e. (102)By Theorem 2.3 in [24] we have e G ∼ = S , therefore π ( X Gal ) is trivial.
In this Section we present our findings. We have considered the following invariants of X Gal : itsfundamental group π ( X Gal ), its Chern numbers c , c , and its signature χ := ( c − c ). .1 The fundamental group of the Galois cover We begin with the fundamental group, for which we were able to deduce, with the exception of asingle case, whether it is trivial.
Theorem 4.1.
The group π ( X Gal ) is not trivial for U , , , U , , , U , , , U , , , U , , , U , , , U , , , U , . The group is trivial in all the other cases, except possibly U , . The case of U , is especially challenging because there are no 1-points, there are no 3-valentvertices in the dual graph not attached to a cycle, and there is no inner 3-point. Thus it is the onlycase for which the techniques presented in the current work are not able to deduce whether π ( X Gal )is trivial. We believe this question can be interesting to those seeking to develop additional algebraictools in the study of Coxeter groups.Note that the technique used in [7] to find the group π ( X Gal ) explicitly is not applicable in higherdegrees. We can therefore formulate the following question.
Question 4.1.
What are the isomorphism classes of the groups π ( X Gal ) , where X degenerates toone of U , , , U , , , U , , , U , , , U , , , U , , , U , , , or U , ? In all those cases we can show that π ( X Gal ) is normally generated by 1 or 2 elements in e G (namely the fork relations). Whenever this group is normally generated by a unique element, basedresults in smaller degrees, we believe all the conjugations of this element commute and thus formulatethe following conjecture: Conjecture 4.1.
The group π ( X Gal ) is free abelian group whenever X can be degenerated to oneof U , , , U , , , U , , , U , , , U , , , U , , , or U , . The Chern numbers and the signature of the Galois cover are additional important topological in-variants. To compute them we use the following proposition.
Proposition 4.2. [21, Proposition 0.2] The Chern classes of X Gal are as follows: ( ) c ( X Gal ) = n !4 ( m − , ( ) c ( X Gal ) = n !(3 − m + d + µ + ρ ) , where n = deg f, m = deg S , µ = number of branch points in S , d = number of nodes in S , and ρ = number of cusps in S . We denote by χ ( X Gal ) the signature of the Galois cover, and we compute it by the formula χ ( X Gal ) = 13 ( c − c ) . Using Proposition 4.2 and the computations appearing in [4, Appendix B], we can obtain theChern numbers of all the Galois covers we consider. heorem 4.3. The Chern numbers and signatures of the Galois covers of the surfaces appear inTable 1.Moreover, all the signatures of degree 6 surfaces with a degeneration that have planar representa-tion are negative and have the form − a · where a ∈ { , . . . , } .Proof. This is an immediate application of Proposition 4.2. For the values of n, m, µ, d and ρ thatappear in this proposition, see [4, Appendix B]. We attach the following table that includes the invariant computed in the paper: Chern numbers,signature and information whether the fundamental group of the Galois cover is trivial or not. X c c χ π ( X Gal ) U , ·
6! 4 · − ·
6! trivial U , , , U , , , U , , , U , , , U , , ·
6! 4 · − ·
6! not trivial U , , ·
6! 3 · − ·
6! not trivial U , , · · −
6! not trivial U , , , U , , · · − ·
6! trivial U , , , U , , ·
6! 5 · − ·
6! trivial U , · · − ·
6! trivial U , , U , , U , · · − ·
6! trivial U , · · − ·
6! not trivial U , , U ·
6! 6 · −
6! trivial U , · · − ·
6! trivial U , ·
6! 7 · − ·
6! trivial U ∪ ·
6! 11 · − ·
6! trivial U , · · − ·
6! trivial U , ·
6! 5 · − ·
6! open question U ∪ , , U ∪ · · −
6! trivial U ∪ , , U ∪ ·
6! 9 · − ·
6! trivial U ·
6! 7 · − ·
6! trivial
Table 1:
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We are grateful to Guo Zhiming for proving the completeness of the following classification.
Theorem A.1.
There are exactly 29 non-isomorphic sextic degenerations that have a planar repre-sentation (see Definition 3.1).Proof.
In order to classify all planar representations, it is equivalent to classify all combinations ofsix plane triangles with common edges. First, we consider a special kind of n-point configuration,which does not change by gluing other triangles. We denote the n-point configuration with thesekinds of n-points by D n (corresponding to degeneration of n-degree surface). Note that the smallest D n is D . First, we classify sextic degenerations by gluing D k with 6 − k triangles. • The largest D n in the configuration is D . There is only one possible configuration (see Figure8), which we denote by U . Figure 8: U . • The largest D n in the configuration is D . In this case, there are only 2-points, except the 5-point, which cannot be changed. Except for the general type (the 5-point quintic degenerationand a plane), only D can be inserted. There is only one possible configuration in each case(see Figure 9), which we denote by U and U ∪ .Figure 9: U (left) and U ∪ (right). • The largest D n in the configuration is D . In this case, there are two cases; one is the config-uration of a degeneration of a union of D with itself, the other one is a union of D with D .There is only one possible degeneration of the first case (see Figure 10), which we denote by U ∪ . igure 10: U ∪ . The second case consists of exactly one D . So there are two possible configurations in thiscase (see Figure 11), which we denote U ∪ , , U ∪ , .Figure 11: U ∪ , (left) and U ∪ , (right). There are also three possible degenerations of a general type (see Figure 12), which we denoteas U , , U , and U , .Figure 12: From left to right: U , , U , , and U , . • The largest D n in the configuration is D . There is only one possible degeneration of D with D (see Figure 13), which we denote U ∪ .Figure 13: U ∪ . Next we consider D and a plane. It is not difficult to see that there are six possible cases (seeFigure 14), which we denote U ,i , i ∈ { , . . . , } . igure 14: Top row (from left to right): U , , U , , and U , .Bottom row (from left to right): U , , U , , and U , . • Now we have the general type, which can be classified from the largest n-point.If n = 7, there is only one possible configuration of a degeneration (see Figure 15), denoted U , . Figure 15: U , . If n = 6, there are three possible configurations of degenerations (see Figure 16), denoted as U , , , U , , , and U , , .Figure 16: From left to right: U , , , U , , , and U , , . If n = 5, there are eight possible configurations of degenerations (see Figures 17, 18, and 19),denoted U , ,i for i = 1 , . . . , igure 17: From left to right: U , , , U , , , and U , , . Figure 18:
From left to right: U , , , U , , , and U , , . Figure 19: U , , (left) and U , , (right). If n = 4, there is only one possible configuration of a degeneration (see Figure 20), denoted U , . Figure 20: U , . Full calculations
In this appendix we provide the full computations we have made. See Section 2 for the notations.
B.1 X degenerates to U , Figure 21:
The arrangement of planes U , . Theorem B.1. If X degenerates to U , , then π ( X Gal ) is trivial.Proof. The branch curve S in CP is an arrangement of five lines. We regenerate each vertex inturn and compute the group G .Vertex 1 is a 1-point that gives rise to braid Z ′ and to the following relation in G :Γ = Γ ′ . (103)Vertex 8 is a 1-point that gives rise to braid Z ′ and to the following relation in G :Γ = Γ ′ . (104)Vertex 2 is a 2-point. The braid monodromy corresponding to this point is: f ∆ = ( Z ′ ) Z
22 2 ′ , · Z
32 2 ′ , . f ∆ gives rise to the following relations: h Γ , Γ i = h Γ ′ , Γ i = h Γ − Γ ′ Γ , Γ i = e (105)Γ ′ = Γ Γ ′ Γ Γ Γ − Γ ′− Γ − . (106)Vertex 3 is a 2-point. The braid monodromy corresponding to this point is: f ∆ = ( Z ′ ) Z
24 4 ′ , · Z
34 4 ′ , . f ∆ gives rise to the following relations: h Γ , Γ i = h Γ ′ , Γ i = h Γ − Γ ′ Γ , Γ i = e (107)Γ ′ = Γ Γ ′ Γ Γ Γ − Γ ′− Γ − . (108) ertex 6 is a 2-point. The braid monodromy corresponding to this point is: f ∆ = ( Z ′ ) Z
21 1 ′ , · Z
31 1 ′ , . f ∆ gives rise to the following relations: h Γ , Γ i = h Γ ′ , Γ i = h Γ − Γ ′ Γ , Γ i = e (109)Γ ′ = Γ Γ ′ Γ Γ Γ − Γ ′− Γ − . (110)Vertex 7 is a 2-point. The braid monodromy corresponding to this point is: f ∆ = ( Z ′ ) Z
23 3 ′ , · Z
33 3 ′ , . f ∆ gives rise to the following relations: h Γ , Γ i = h Γ ′ , Γ i = h Γ − Γ ′ Γ , Γ i = e (111)Γ ′ = Γ Γ ′ Γ Γ Γ − Γ ′− Γ − . (112)We also have the following parasitic and projective relations:[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (113)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (114)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (115)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (116)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (117)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (118)Γ ′ Γ Γ ′ Γ Γ ′ Γ Γ ′ Γ Γ ′ Γ = e. (119)By Lemma 3.3 we get in e G that Γ = Γ ′ , Γ ′ = Γ and Γ = Γ ′ .Thus e G is generated by { Γ i | i = 1 , . . . , } modulo the following relations:Γ = Γ = Γ = Γ = Γ = e (120) h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = e (121)[Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = e. (122)The generators of e G satisfy the same relations as the transpositions in S , therefore e G ∼ = S and π ( X Gal ) is trivial.In this case, the number of branch points is 6, the number of cusps is 12, the number of nodes is24 and deg S = 10. So the first Chern number is c = 4 · c = 4 · χ = − · .2 X degenerates to U , ,
12 3 45 6 78 123 4 5
Figure 22:
The arrangement of planes U , , . Theorem B.2. If X degenerates to U , , , then π ( X Gal ) is not trivial.Proof. The branch curve S in CP is an arrangement of five lines. We regenerate each vertex inturn and compute the group G .Vertex 2 is a 1-point that gives rise to braid Z ′ and to the following relation in G :Γ = Γ ′ . (123)Vertex 4 is a 1-point that gives rise to braid Z ′ and to the following relation in G :Γ = Γ ′ . (124)Vertex 5 is a 2-point. The braid monodromy corresponding to this point is: f ∆ = ( Z ′ ) Z
22 2 ′ , · Z
32 2 ′ , . f ∆ gives rise to the following relations: h Γ , Γ i = h Γ ′ , Γ i = h Γ − Γ ′ Γ , Γ i = e (125)Γ ′ = Γ Γ ′ Γ Γ Γ − Γ ′− Γ − . (126)Vertex 3 is an outer 3-point. The braid monodromy corresponding to this 3-point is: f ∆ = Z
21 1 ′ , ′ · Z ′ , ′ · ( Z ′ ) Z ′ , ′ · ( Z
32 2 ′ , ) Z ′ , ′ · ( Z ′ ) Z
22 2 ′ , Z ′ , ′ . f ∆ thus gives rise to the following relations: h Γ ′ , Γ i = h Γ ′ , Γ ′ i = h Γ ′ , Γ − Γ ′ Γ i = e (127) = Γ ′ Γ Γ ′ Γ − Γ ′ − (128) h Γ , Γ ′ Γ Γ ′ Γ Γ ′ − Γ − Γ ′ − i = h Γ , Γ ′ Γ Γ ′ Γ ′ Γ ′ − Γ − Γ ′ − i == h Γ , Γ ′ Γ Γ ′ Γ − Γ ′ Γ Γ ′ − Γ − Γ ′ − i = e (129)Γ ′ = Γ Γ ′ Γ Γ ′ Γ ′ Γ Γ ′ − Γ − Γ ′ − Γ Γ ′ Γ Γ ′ Γ − Γ ′ − Γ ′ − Γ − Γ ′ − Γ − (130)[Γ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ] = [Γ ′ , Γ ′ ] = e. (131)Vertex 6 is an outer 3-point. The braid monodromy corresponding to this 3-point is: f ∆ = Z
23 3 ′ , ′ · Z ′ , ′ · ( Z ′ ) Z ′ , ′ · ( Z
34 4 ′ , ) Z ′ , ′ · ( Z ′ ) Z
24 4 ′ , Z ′ , ′ . f ∆ thus gives rise to the following relations: h Γ ′ , Γ i = h Γ ′ , Γ ′ i = h Γ ′ , Γ − Γ ′ Γ i = e (132)Γ = Γ ′ Γ Γ ′ Γ − Γ ′ − (133) h Γ , Γ ′ Γ Γ ′ Γ Γ ′ − Γ − Γ ′ − i = h Γ , Γ ′ Γ Γ ′ Γ ′ Γ ′ − Γ − Γ ′ − i == h Γ , Γ ′ Γ Γ ′ Γ − Γ ′ Γ Γ ′ − Γ − Γ ′ − i = e (134)Γ ′ = Γ Γ ′ Γ Γ ′ Γ ′ Γ Γ ′ − Γ − Γ ′ − Γ Γ ′ Γ Γ ′ Γ − Γ ′ − Γ ′ − Γ − Γ ′ − Γ − (135)[Γ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ] = [Γ ′ , Γ ′ ] = e. (136)We also have the following parasitic and projective relations:[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (137)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (138)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (139)Γ ′ Γ Γ ′ Γ Γ ′ Γ Γ ′ Γ Γ ′ Γ = e. (140)By Lemma 3.5 we get Γ = Γ ′ and Γ = Γ ′ . Then by Lemma 3.3, Γ = Γ ′ .Thus e G is generated by { Γ i | i = 1 , . . . , } modulo the following relations:Γ = Γ = Γ = Γ = Γ = e (141) h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = e (142)[Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = e. (143)By Lemma 3.2, π ( X Gal ) is not trivial.In this case, the number of branch points is 7, the number of cusps is 15, the number of nodes is20 and deg S = 10. So the first Chern number is c = 4 · c = 4 · χ = − · .3 X degenerates to U , , Figure 23:
The arrangement of planes U , , . Theorem B.3. If X degenerates to U , , , then π ( X Gal ) is not trivial.Proof. The branch curve S in CP is an arrangement of five lines. We regenerate each vertex inturn and compute the group G .Vertex 1 is a 1-point that gives rise to braid Z ′ and to the following relation in G :Γ = Γ ′ . (144)Vertex 3 is a 1-point that gives rise to braid Z ′ and to the following relation in G :Γ = Γ ′ . (145)Vertex 2 is a 2-point. The braid monodromy corresponding to this point is: f ∆ = ( Z ′ ) Z
22 2 ′ , · Z
32 2 ′ , . f ∆ gives rise to the following relations: h Γ , Γ i = h Γ ′ , Γ i = h Γ − Γ ′ Γ , Γ i = e (146)Γ ′ = Γ Γ ′ Γ Γ Γ − Γ ′− Γ − . (147)Vertex 6 is an outer 3-point. The braid monodromy corresponding to this 3-point is: f ∆ = Z
21 1 ′ , ′ · Z ′ , ′ · ( Z ′ ) Z ′ , ′ · ( Z
32 2 ′ , ) Z ′ , ′ · ( Z ′ ) Z
22 2 ′ , Z ′ , ′ . f ∆ thus gives rise to the following relations: h Γ ′ , Γ i = h Γ ′ , Γ ′ i = h Γ ′ , Γ − Γ ′ Γ i = e (148)Γ = Γ ′ Γ Γ ′ Γ − Γ ′ − (149) Γ , Γ ′ Γ Γ ′ Γ Γ ′ − Γ − Γ ′ − i = h Γ , Γ ′ Γ Γ ′ Γ ′ Γ ′ − Γ − Γ ′ − i == h Γ , Γ ′ Γ Γ ′ Γ − Γ ′ Γ Γ ′ − Γ − Γ ′ − i = e (150)Γ ′ = Γ Γ ′ Γ Γ ′ Γ ′ Γ Γ ′ − Γ − Γ ′ − Γ Γ ′ Γ Γ ′ Γ − Γ ′ − Γ ′ − Γ − Γ ′ − Γ − (151)[Γ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ] = [Γ ′ , Γ ′ ] = e. (152)Vertex 7 is an outer 3-point. The braid monodromy corresponding to this 3-point is: f ∆ = Z
23 3 ′ , ′ · Z ′ , ′ · ( Z ′ ) Z ′ , ′ · ( Z
34 4 ′ , ) Z ′ , ′ · ( Z ′ ) Z
24 4 ′ , Z ′ , ′ . f ∆ thus gives rise to the following relations: h Γ ′ , Γ i = h Γ ′ , Γ ′ i = h Γ ′ , Γ − Γ ′ Γ i = e (153)Γ = Γ ′ Γ Γ ′ Γ − Γ ′ − (154) h Γ , Γ ′ Γ Γ ′ Γ Γ ′ − Γ − Γ ′ − i = h Γ , Γ ′ Γ Γ ′ Γ ′ Γ ′ − Γ − Γ ′ − i == h Γ , Γ ′ Γ Γ ′ Γ − Γ ′ Γ Γ ′ − Γ − Γ ′ − i = e (155)Γ ′ = Γ Γ ′ Γ Γ ′ Γ ′ Γ Γ ′ − Γ − Γ ′ − Γ Γ ′ Γ Γ ′ Γ − Γ ′ − Γ ′ − Γ − Γ ′ − Γ − (156)[Γ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ] = [Γ ′ , Γ ′ ] = e. (157)We also have the following parasitic and projective relations:[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (158)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (159)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (160)Γ ′ Γ Γ ′ Γ Γ ′ Γ Γ ′ Γ Γ ′ Γ = e. (161)By Lemma 3.5 we get Γ = Γ ′ and Γ = Γ ′ . Then by Lemma 3.3, Γ = Γ ′ .Thus e G is generated by { Γ i | i = 1 , . . . , } modulo the following relations:Γ = Γ = Γ = Γ = Γ = e (162) h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = e (163)[Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = e. (164)By Lemma 3.2, π ( X Gal ) is not trivial.In this case, the number of branch points is 7, the number of cusps is 15, the number of nodes is20 and deg S = 10. So the first Chern number is c = 4 · c = 4 · χ = − · .4 X degenerates to U , ,
12 3 45 6 78 123 4 5
Figure 24:
The arrangement of planes U , , . Theorem B.4. If X degenerates to U , , , then π ( X Gal ) is not trivial.Proof. See Theorem 3.6.In this case, the number of branch points is 7, the number of cusps is 15, the number of nodes is20 and deg S = 10. So the first Chern number is c = 4 · c = 4 · χ = − · B.5 X degenerates to U , ,
12 3 45 6 7 812 3 4 5
Figure 25:
The arrangement of planes U , , . heorem B.5. If X degenerates to U , , , then π ( X Gal ) is not trivial.Proof. The branch curve S in CP is an arrangement of five lines. We regenerate each vertex inturn and compute the group G .Vertex 1 is a 2-point. The braid monodromy corresponding to this point is: f ∆ = ( Z ′ ) Z
21 1 ′ , · Z
31 1 ′ , . f ∆ gives rise to the following relations: h Γ , Γ i = h Γ ′ , Γ i = h Γ − Γ ′ Γ , Γ i = e (165)Γ ′ = Γ Γ ′ Γ Γ Γ − Γ ′− Γ − . (166)Vertex 8 is a 2-point. The braid monodromy corresponding to this point is: f ∆ = ( Z ′ ) Z
24 4 ′ , · Z
34 4 ′ , . f ∆ gives rise to the following relations: h Γ , Γ i = h Γ ′ , Γ i = h Γ − Γ ′ Γ , Γ i = e (167)Γ ′ = Γ Γ ′ Γ Γ Γ − Γ ′− Γ − . (168)Vertex 3 is an outer 3-point. The braid monodromy corresponding to this 3-point is: f ∆ = Z
22 2 ′ , ′ · Z ′ , ′ · ( Z ′ ) Z ′ , ′ · ( Z
33 3 ′ , ) Z ′ , ′ · ( Z ′ ) Z
23 3 ′ , Z ′ , ′ . f ∆ thus gives rise to the following relations: h Γ ′ , Γ i = h Γ ′ , Γ ′ i = h Γ ′ , Γ − Γ ′ Γ i = e (169)Γ = Γ ′ Γ Γ ′ Γ − Γ ′ − (170) h Γ , Γ ′ Γ Γ ′ Γ Γ ′ − Γ − Γ ′ − i = h Γ , Γ ′ Γ Γ ′ Γ ′ Γ ′ − Γ − Γ ′ − i == h Γ , Γ ′ Γ Γ ′ Γ − Γ ′ Γ Γ ′ − Γ − Γ ′ − i = e (171)Γ ′ = Γ Γ ′ Γ Γ ′ Γ ′ Γ Γ ′ − Γ − Γ ′ − Γ Γ ′ Γ Γ ′ Γ − Γ ′ − Γ ′ − Γ − Γ ′ − Γ − (172)[Γ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ] = [Γ ′ , Γ ′ ] = e. (173)Vertex 6 is an outer 3-point. The braid monodromy corresponding to this 3-point is: f ∆ = Z
21 1 ′ , ′ · Z ′ , ′ · ( Z ′ ) Z ′ , ′ · ( Z
33 3 ′ , ) Z ′ , ′ · ( Z ′ ) Z
23 3 ′ , Z ′ , ′ . f ∆ thus gives rise to the following relations: h Γ ′ , Γ i = h Γ ′ , Γ ′ i = h Γ ′ , Γ − Γ ′ Γ i = e (174) = Γ ′ Γ Γ ′ Γ − Γ ′ − (175) h Γ , Γ ′ Γ Γ ′ Γ Γ ′ − Γ − Γ ′ − i = h Γ , Γ ′ Γ Γ ′ Γ ′ Γ ′ − Γ − Γ ′ − i == h Γ , Γ ′ Γ Γ ′ Γ − Γ ′ Γ Γ ′ − Γ − Γ ′ − i = e (176)Γ ′ = Γ Γ ′ Γ Γ ′ Γ ′ Γ Γ ′ − Γ − Γ ′ − Γ Γ ′ Γ Γ ′ Γ − Γ ′ − Γ ′ − Γ − Γ ′ − Γ − (177)[Γ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ] = [Γ ′ , Γ ′ ] = e. (178)We also have the following parasitic and projective relations:[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (179)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (180)Γ ′ Γ Γ ′ Γ Γ ′ Γ Γ ′ Γ Γ ′ Γ = e. (181)We compute the quotient e G/ h Γ i = Γ ′ i i :Γ = Γ = Γ = Γ = Γ = e (182) h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = e (183)[Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = e. (184)By Lemma 3.2, π ( X Gal ) is not trivial.In this case, the number of branch points is 6, the number of cusps is 18, the number of nodes is16 and deg S = 10. So the first Chern number is c = 4 · c = 3 · χ = − · B.6 X degenerates to U , , Figure 26:
The arrangement of planes U , , . Theorem B.6. If X degenerates to U , , , then π ( X Gal ) is not trivial. roof. The branch curve S in CP is an arrangement of five lines. We regenerate each vertex inturn and compute the group G .Vertex 1 is a 1-point that gives rise to braid Z ′ and to the following relation in G :Γ = Γ ′ . (185)Vertex 4 is a 2-point. The braid monodromy corresponding to this point is: f ∆ = ( Z ′ ) Z
24 4 ′ , · Z
34 4 ′ , . f ∆ gives rise to the following relations: h Γ , Γ i = h Γ ′ , Γ i = h Γ − Γ ′ Γ , Γ i = e (186)Γ ′ = Γ Γ ′ Γ Γ Γ − Γ ′− Γ − . (187)Vertex 7 is a 2-point. The braid monodromy corresponding to this point is: f ∆ = ( Z ′ ) Z
21 1 ′ , · Z
31 1 ′ , . f ∆ gives rise to the following relations: h Γ , Γ i = h Γ ′ , Γ i = h Γ − Γ ′ Γ , Γ i = e (188)Γ ′ = Γ Γ ′ Γ Γ Γ − Γ ′− Γ − . (189)Vertex 8 is a 2-point. The braid monodromy corresponding to this point is: f ∆ = ( Z ′ ) Z
23 3 ′ , · Z
33 3 ′ , . f ∆ gives rise to the following relations: h Γ , Γ i = h Γ ′ , Γ i = h Γ − Γ ′ Γ , Γ i = e (190)Γ ′ = Γ Γ ′ Γ Γ Γ − Γ ′− Γ − . (191)Vertex 2 is an outer 3-point. The braid monodromy corresponding to this 3-point is: f ∆ = Z
22 2 ′ , ′ · Z ′ , ′ · ( Z ′ ) Z ′ , ′ · ( Z
33 3 ′ , ) Z ′ , ′ · ( Z ′ ) Z
23 3 ′ , Z ′ , ′ . f ∆ thus gives rise to the following relations: h Γ ′ , Γ i = h Γ ′ , Γ ′ i = h Γ ′ , Γ − Γ ′ Γ i = e (192)Γ = Γ ′ Γ Γ ′ Γ − Γ ′ − (193) h Γ , Γ ′ Γ Γ ′ Γ Γ ′ − Γ − Γ ′ − i = h Γ , Γ ′ Γ Γ ′ Γ ′ Γ ′ − Γ − Γ ′ − i == h Γ , Γ ′ Γ Γ ′ Γ − Γ ′ Γ Γ ′ − Γ − Γ ′ − i = e (194) ′ = Γ Γ ′ Γ Γ ′ Γ ′ Γ Γ ′ − Γ − Γ ′ − Γ Γ ′ Γ Γ ′ Γ − Γ ′ − Γ ′ − Γ − Γ ′ − Γ − (195)[Γ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ] = [Γ ′ , Γ ′ ] = e. (196)We also have the following parasitic and projective relations:[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (197)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (198)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (199)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (200)Γ ′ Γ Γ ′ Γ Γ ′ Γ Γ ′ Γ Γ ′ Γ = e. (201)By Lemma 3.3, Γ = Γ ′ . Then by Lemma 3.5 we get Γ = Γ ′ and Γ = Γ ′ . Again by Lemma3.3, we get Γ = Γ ′ .Thus e G is generated by { Γ i | i = 1 , . . . , } modulo the following relations:Γ = Γ = Γ = Γ = Γ = e (202) h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = e (203)[Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = e. (204)By Lemma 3.2, π ( X Gal ) is not trivial.In this case, the number of branch points is 6, the number of cusps is 15, the number of nodes is20 and deg S = 10. So the first Chern number is c = 4 · c = · χ = − B.7 X degenerates to U , , Figure 27:
The arrangement of planes U , , . Theorem B.7. If X degenerates to U , , , then π ( X Gal ) is trivial. roof. The branch curve S in CP is an arrangement of five lines. We regenerate each vertex inturn and compute the group G .Vertex 1 is a 1-point that gives rise to braid Z ′ and to the following relation in G :Γ = Γ ′ . (205)Vertex 3 is a 1-point that gives rise to braid Z ′ and to the following relation in G :Γ = Γ ′ . (206)Vertex 4 is a 1-point that gives rise to braid Z ′ and to the following relation in G :Γ = Γ ′ . (207)Vertex 2 is a 2-point. The braid monodromy corresponding to this point is: f ∆ = ( Z ′ ) Z
22 2 ′ , · Z
32 2 ′ , . f ∆ gives rise to the following relations: h Γ , Γ i = h Γ ′ , Γ i = h Γ − Γ ′ Γ , Γ i = e (208)Γ ′ = Γ Γ ′ Γ Γ Γ − Γ ′− Γ − . (209)Vertex 6 is a 2-point. The braid monodromy corresponding to this point is: f ∆ = ( Z ′ ) Z
21 1 ′ , · Z
31 1 ′ , . f ∆ gives rise to the following relations: h Γ , Γ i = h Γ ′ , Γ i = h Γ − Γ ′ Γ , Γ i = e (210)Γ ′ = Γ Γ ′ Γ Γ Γ − Γ ′− Γ − . (211)Vertex 7 is an outer 3-point. The braid monodromy corresponding to this 3-point is: f ∆ = Z
23 3 ′ , ′ · Z ′ , ′ · ( Z ′ ) Z ′ , ′ · ( Z
34 4 ′ , ) Z ′ , ′ · ( Z ′ ) Z
24 4 ′ , Z ′ , ′ . f ∆ thus gives rise to the following relations: h Γ ′ , Γ i = h Γ ′ , Γ ′ i = h Γ ′ , Γ − Γ ′ Γ i = e (212)Γ = Γ ′ Γ Γ ′ Γ − Γ ′ − (213) h Γ , Γ ′ Γ Γ ′ Γ Γ ′ − Γ − Γ ′ − i = h Γ , Γ ′ Γ Γ ′ Γ ′ Γ ′ − Γ − Γ ′ − i == h Γ , Γ ′ Γ Γ ′ Γ − Γ ′ Γ Γ ′ − Γ − Γ ′ − i = e (214)Γ ′ = Γ Γ ′ Γ Γ ′ Γ ′ Γ Γ ′ − Γ − Γ ′ − Γ Γ ′ Γ Γ ′ Γ − Γ ′ − Γ ′ − Γ − Γ ′ − Γ − (215) Γ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ] = [Γ ′ , Γ ′ ] = e. (216)We also have the following parasitic and projective relations:[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (217)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (218)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (219)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (220)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (221)Γ ′ Γ Γ ′ Γ Γ ′ Γ Γ ′ Γ Γ ′ Γ = e. (222)By Lemma 3.3, we get Γ = Γ ′ and Γ = Γ ′ . e G is thus generated by { Γ i | i = 1 , . . . , } with the following relations:Γ = Γ = Γ = Γ = Γ = e (223) h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = e (224)[Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = e. (225)Thus, e G ∼ = S , therefore π ( X Gal ) is trivial.In this case, the number of branch points is 7, the number of cusps is 12, the number of nodes is24 and deg S = 10. So the first Chern number is c = 4 · c = · χ = − · B.8 X degenerates to U , , Figure 28:
The arrangement of planes U , , . Theorem B.8. If X degenerates to U , , , then π ( X Gal ) is trivial. roof. The branch curve S in CP is an arrangement of five lines. We regenerate each vertex inturn and compute the group G .Vertex 1 is a 1-point that gives rise to braid Z ′ and to the following relation in G :Γ = Γ ′ . (226)Vertex 2 is a 1-point that gives rise to braid Z ′ and to the following relation in G :Γ = Γ ′ . (227)Vertex 7 is a 1-point that gives rise to braid Z ′ and to the following relation in G :Γ = Γ ′ . (228)Vertex 8 is a 1-point that gives rise to braid Z ′ and to the following relation in G :Γ = Γ ′ . (229)Vertex 3 is an outer 3-point. The braid monodromy corresponding to this 3-point is: f ∆ = Z
23 3 ′ , ′ · Z ′ , ′ · ( Z ′ ) Z ′ , ′ · ( Z
34 4 ′ , ) Z ′ , ′ · ( Z ′ ) Z
24 4 ′ , Z ′ , ′ . f ∆ thus gives rise to the following relations: h Γ ′ , Γ i = h Γ ′ , Γ ′ i = h Γ ′ , Γ − Γ ′ Γ i = e (230)Γ = Γ ′ Γ Γ ′ Γ − Γ ′ − (231) h Γ , Γ ′ Γ Γ ′ Γ Γ ′ − Γ − Γ ′ − i = h Γ , Γ ′ Γ Γ ′ Γ ′ Γ ′ − Γ − Γ ′ − i == h Γ , Γ ′ Γ Γ ′ Γ − Γ ′ Γ Γ ′ − Γ − Γ ′ − i = e (232)Γ ′ = Γ Γ ′ Γ Γ ′ Γ ′ Γ Γ ′ − Γ − Γ ′ − Γ Γ ′ Γ Γ ′ Γ − Γ ′ − Γ ′ − Γ − Γ ′ − Γ − (233)[Γ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ] = [Γ ′ , Γ ′ ] = e. (234)Vertex 6 is an outer 3-point. The braid monodromy corresponding to this 3-point is: f ∆ = Z
21 1 ′ , ′ · Z ′ , ′ · ( Z ′ ) Z ′ , ′ · ( Z
32 2 ′ , ) Z ′ , ′ · ( Z ′ ) Z
22 2 ′ , Z ′ , ′ . f ∆ thus gives rise to the following relations: h Γ ′ , Γ i = h Γ ′ , Γ ′ i = h Γ ′ , Γ − Γ ′ Γ i = e (235)Γ = Γ ′ Γ Γ ′ Γ − Γ ′ − (236) h Γ , Γ ′ Γ Γ ′ Γ Γ ′ − Γ − Γ ′ − i = h Γ , Γ ′ Γ Γ ′ Γ ′ Γ ′ − Γ − Γ ′ − i == h Γ , Γ ′ Γ Γ ′ Γ − Γ ′ Γ Γ ′ − Γ − Γ ′ − i = e (237)Γ ′ = Γ Γ ′ Γ Γ ′ Γ ′ Γ Γ ′ − Γ − Γ ′ − Γ Γ ′ Γ Γ ′ Γ − Γ ′ − Γ ′ − Γ − Γ ′ − Γ − (238) Γ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ] = [Γ ′ , Γ ′ ] = e. (239)We also have the following parasitic and projective relations:[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (240)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (241)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (242)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (243)Γ ′ Γ Γ ′ Γ Γ ′ Γ Γ ′ Γ Γ ′ Γ = e. (244)By Lemma 3.5, Γ = Γ ′ . e G is thus generated by { Γ i | i = 1 , . . . , } with the following relations:Γ = Γ = Γ = Γ = Γ = e (245) h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = e (246)[Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = e. (247)Thus, e G ∼ = S , therefore π ( X Gal ) is trivial.In this case, the number of branch points is 8, the number of cusps is 12, the number of nodes is24 and deg S = 10. So the first Chern number is c = 4 · c = 5 · χ = − · B.9 X degenerates to U , , Figure 29:
The arrangement of planes U , , . Theorem B.9. If X degenerates to U , , , then π ( X Gal ) is trivial. roof. The branch curve S in CP is an arrangement of five lines. We regenerate each vertex inturn and compute the group G .Vertex 2 is a 1-point that gives rise to braid Z ′ and to the following relation in G :Γ = Γ ′ . (248)Vertex 5 is a 1-point that gives rise to braid Z ′ and to the following relation in G :Γ = Γ ′ . (249)Vertex 6 is a 1-point that gives rise to braid Z ′ and to the following relation in G :Γ = Γ ′ . (250)Vertex 3 is a 2-point. The braid monodromy corresponding to this point is: f ∆ = ( Z ′ ) Z
21 1 ′ , · Z
31 1 ′ , . f ∆ gives rise to the following relations: h Γ , Γ i = h Γ ′ , Γ i = h Γ − Γ ′ Γ , Γ i = e (251)Γ ′ = Γ Γ ′ Γ Γ Γ − Γ ′− Γ − . (252)Vertex 7 is a 2-point. The braid monodromy corresponding to this point is: f ∆ = ( Z ′ ) Z
24 4 ′ , · Z
34 4 ′ , . f ∆ gives rise to the following relations: h Γ , Γ i = h Γ ′ , Γ i = h Γ − Γ ′ Γ , Γ i = e (253)Γ ′ = Γ Γ ′ Γ Γ Γ − Γ ′− Γ − . (254)Vertex 4 is an outer 3-point. The braid monodromy corresponding to this 3-point is: f ∆ = Z
22 2 ′ , ′ · Z ′ , ′ · ( Z ′ ) Z ′ , ′ · ( Z
33 3 ′ , ) Z ′ , ′ · ( Z ′ ) Z
23 3 ′ , Z ′ , ′ . f ∆ thus gives rise to the following relations: h Γ ′ , Γ i = h Γ ′ , Γ ′ i = h Γ ′ , Γ − Γ ′ Γ i = e (255)Γ = Γ ′ Γ Γ ′ Γ − Γ ′ − (256) h Γ , Γ ′ Γ Γ ′ Γ Γ ′ − Γ − Γ ′ − i = h Γ , Γ ′ Γ Γ ′ Γ ′ Γ ′ − Γ − Γ ′ − i == h Γ , Γ ′ Γ Γ ′ Γ − Γ ′ Γ Γ ′ − Γ − Γ ′ − i = e (257)Γ ′ = Γ Γ ′ Γ Γ ′ Γ ′ Γ Γ ′ − Γ − Γ ′ − Γ Γ ′ Γ Γ ′ Γ − Γ ′ − Γ ′ − Γ − Γ ′ − Γ − (258) Γ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ] = [Γ ′ , Γ ′ ] = e. (259)We also have the following parasitic and projective relations:[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (260)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (261)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (262)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (263)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (264)Γ ′ Γ Γ ′ Γ Γ ′ Γ Γ ′ Γ Γ ′ Γ = e. (265)By Lemma 3.3, we get Γ = Γ ′ and Γ = Γ ′ . e G is thus generated by { Γ i | i = 1 , . . . , } with the following relations:Γ = Γ = Γ = Γ = Γ = e (266) h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = e (267)[Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = e. (268)Thus, e G ∼ = S , therefore π ( X Gal ) is trivial.In this case, the number of branch points is 7, the number of cusps is 12, the number of nodes is24 and deg S = 10. So the first Chern number is c = 4 · c = · χ = − · B.10 X degenerates to U , , Figure 30:
The arrangement of planes U , , . Theorem B.10. If X degenerates to U , , , then π ( X Gal ) is trivial. roof. See Theorem 3.7.In this case, the number of branch points is 8, the number of cusps is 12, the number of nodes is24 and deg S = 10. So the first Chern number is c = 4 · c = 5 · χ = − · B.11 X degenerates to U , , Figure 31:
The arrangement of planes U , , . Theorem B.11. If X degenerates to U , , , then π ( X Gal ) is not trivial.Proof. The branch curve S in CP is an arrangement of five lines. We regenerate each vertex inturn and compute the group G .Vertex 3 is a 1-point that gives rise to braid Z ′ and to the following relation in G :Γ = Γ ′ . (269)Vertex 6 is a 1-point that gives rise to braid Z ′ and to the following relation in G :Γ = Γ ′ . (270)Vertex 5 is a 2-point. The braid monodromy corresponding to this point is: f ∆ = ( Z ′ ) Z
24 4 ′ , · Z
34 4 ′ , . f ∆ gives rise to the following relations: h Γ , Γ i = h Γ ′ , Γ i = h Γ − Γ ′ Γ , Γ i = e (271) ′ = Γ Γ ′ Γ Γ Γ − Γ ′− Γ − . (272)Vertex 7 is a 2-point. The braid monodromy corresponding to this point is: f ∆ = ( Z ′ ) Z
23 3 ′ , · Z
33 3 ′ , . f ∆ gives rise to the following relations: h Γ , Γ i = h Γ ′ , Γ i = h Γ − Γ ′ Γ , Γ i = e (273)Γ ′ = Γ Γ ′ Γ Γ Γ − Γ ′− Γ − . (274)Vertex 4 is a 4-point. Its braid monodromy is e ∆ = Z
33 3 ′ , · ( Z
21 1 ′ , ) Z −
22 2 ′ , · Z
22 2 ′ , · ¯ Z
21 1 ′ , ′ · ¯ Z
22 2 ′ , ′ · ( Z ′ ) Z
23 3 ′ , · Z ′ , ′ · ( Z ′ ) Z ′ , ′ · ( Z
32 2 ′ , ) Z ′ , ′ Z
23 3 ′ , · ( Z ′ ) Z
22 2 ′ , Z ′ , ′ Z
23 3 ′ , · ( Z
21 1 ′ , ′ ) Z
23 3 ′ , . It gives rise to the following relations: h Γ , Γ i = h Γ ′ , Γ i = h Γ − Γ ′ Γ , Γ i = e (275)[Γ ′ Γ Γ Γ − Γ ′− , Γ ] = [Γ ′ Γ Γ ′ Γ − Γ ′− , Γ ] = e (276)[Γ , Γ ] = [Γ ′ , Γ ] = e (277)[Γ ′ Γ Γ ′ Γ Γ Γ − Γ ′− Γ − Γ ′− , Γ − Γ ′ Γ ] = [Γ ′ Γ Γ ′ Γ Γ ′ Γ − Γ ′− Γ − Γ ′− , Γ − Γ ′ Γ ] = e (278)[Γ ′ Γ Γ Γ − Γ ′− , Γ − Γ ′ Γ ] = [Γ ′ Γ Γ ′ Γ − Γ ′− , Γ − Γ ′ Γ ] = e (279)Γ Γ ′ Γ Γ Γ − Γ ′− Γ − = Γ ′ (280) h Γ ′ , Γ i = h Γ ′ , Γ ′ i = h Γ ′ , Γ − Γ ′ Γ i = e (281)Γ = Γ ′ Γ Γ ′ Γ − Γ ′− (282) h Γ ′ Γ Γ ′ Γ Γ ′− Γ − Γ ′− , Γ Γ Γ − i = h Γ ′ Γ Γ ′ Γ ′ Γ ′− Γ − Γ ′− , Γ Γ Γ − i == h Γ ′ Γ Γ ′ Γ − Γ ′ Γ Γ ′− Γ − Γ ′− , Γ Γ Γ − i = e (283)Γ ′ Γ Γ ′ Γ − Γ ′− Γ ′− Γ − Γ ′− Γ Γ − Γ ′ Γ Γ − Γ ′ Γ Γ ′ Γ ′ Γ Γ ′− Γ − Γ ′− = Γ Γ Γ − (284)[Γ , Γ Γ Γ − ] = [Γ ′ , Γ Γ Γ − ] = [Γ , Γ Γ ′ Γ − ] = [Γ ′ , Γ Γ ′ Γ − ] = e. (285)We also have the following parasitic and projective relations:[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (286)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (287)Γ ′ Γ Γ ′ Γ Γ ′ Γ Γ ′ Γ Γ ′ Γ = e. (288)Substituting (269) and (270) in (284), we get Γ = Γ ′ . By Lemma 3.3 , we have Γ = Γ ′ andΓ = Γ ′ . G is thus generated by { Γ i | i = 1 , . . . , } with the following relations:Γ = Γ = Γ = Γ = Γ = e (289) h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = e (290)[Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = e. (291)By Lemma 3.2, π ( X Gal ) is not trivial.In this case, the number of branch points is 7, the number of cusps is 15, the number of nodes is20 and deg S = 10. So the first Chern number is c = 4 · c = 4 · χ = − · B.12 X degenerates to U , , Figure 32:
The arrangement of planes U , , . Theorem B.12. If X degenerates to U , , , then π ( X Gal ) is not trivial.Proof. The branch curve S in CP is an arrangement of five lines. We regenerate each vertex inturn and compute the group G .Vertex 3 is a 1-point that gives rise to braid Z ′ and to the following relation in G :Γ = Γ ′ . (292)Vertex 5 is a 1-point that gives rise to braid Z ′ and to the following relation in G :Γ = Γ ′ . (293) ertex 6 is a 2-point. The braid monodromy corresponding to this point is: f ∆ = ( Z ′ ) Z
22 2 ′ , · Z
32 2 ′ , . f ∆ gives rise to the following relations: h Γ , Γ i = h Γ ′ , Γ i = h Γ − Γ ′ Γ , Γ i = e (294)Γ ′ = Γ Γ ′ Γ Γ Γ − Γ ′− Γ − . (295)Vertex 7 is a 2-point. The braid monodromy corresponding to this point is: f ∆ = ( Z ′ ) Z
23 3 ′ , · Z
33 3 ′ , . f ∆ gives rise to the following relations: h Γ , Γ i = h Γ ′ , Γ i = h Γ − Γ ′ Γ , Γ i = e (296)Γ ′ = Γ Γ ′ Γ Γ Γ − Γ ′− Γ − . (297)Vertex 4 is a 4-point. Its braid monodromy is e ∆ = Z
33 3 ′ , · ( Z
21 1 ′ , ) Z −
22 2 ′ , · Z
22 2 ′ , · ¯ Z
21 1 ′ , ′ · ¯ Z
22 2 ′ , ′ · ( Z ′ ) Z
23 3 ′ , · Z ′ , ′ · ( Z ′ ) Z ′ , ′ · ( Z
32 2 ′ , ) Z ′ , ′ Z
23 3 ′ , · ( Z ′ ) Z
22 2 ′ , Z ′ , ′ Z
23 3 ′ , · ( Z
21 1 ′ , ′ ) Z
23 3 ′ , . It gives rise to the following relations: h Γ , Γ i = h Γ ′ , Γ i = h Γ − Γ ′ Γ , Γ i = e (298)[Γ ′ Γ Γ Γ − Γ ′− , Γ ] = [Γ ′ Γ Γ ′ Γ − Γ ′− , Γ ] = e (299)[Γ , Γ ] = [Γ ′ , Γ ] = e (300)[Γ ′ Γ Γ ′ Γ Γ Γ − Γ ′− Γ − Γ ′− , Γ − Γ ′ Γ ] = [Γ ′ Γ Γ ′ Γ Γ ′ Γ − Γ ′− Γ − Γ ′− , Γ − Γ ′ Γ ] = e (301)[Γ ′ Γ Γ Γ − Γ ′− , Γ − Γ ′ Γ ] = [Γ ′ Γ Γ ′ Γ − Γ ′− , Γ − Γ ′ Γ ] = e (302)Γ Γ ′ Γ Γ Γ − Γ ′− Γ − = Γ ′ (303) h Γ ′ , Γ i = h Γ ′ , Γ ′ i = h Γ ′ , Γ − Γ ′ Γ i = e (304)Γ = Γ ′ Γ Γ ′ Γ − Γ ′− (305) h Γ ′ Γ Γ ′ Γ Γ ′− Γ − Γ ′− , Γ Γ Γ − i = h Γ ′ Γ Γ ′ Γ ′ Γ ′− Γ − Γ ′− , Γ Γ Γ − i == h Γ ′ Γ Γ ′ Γ − Γ ′ Γ Γ ′− Γ − Γ ′− , Γ Γ Γ − i = e (306)Γ ′ Γ Γ ′ Γ − Γ ′− Γ ′− Γ − Γ ′− Γ Γ − Γ ′ Γ Γ − Γ ′ Γ Γ ′ Γ ′ Γ Γ ′− Γ − Γ ′− = Γ Γ Γ − (307)[Γ , Γ Γ Γ − ] = [Γ ′ , Γ Γ Γ − ] = [Γ , Γ Γ ′ Γ − ] = [Γ ′ , Γ Γ ′ Γ − ] = e. (308) e also have the following parasitic and projective relations:[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (309)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (310)Γ ′ Γ Γ ′ Γ Γ ′ Γ Γ ′ Γ Γ ′ Γ = e. (311)Using (293) and (298) in (303), we get Γ = Γ ′ . We use also (292) and (304) in (305), we getΓ = Γ ′ . By Lemma 3.3 , we have Γ = Γ ′ . e G is thus generated by { Γ i | i = 1 , . . . , } with the following relations:Γ = Γ = Γ = Γ = Γ = e (312) h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = e, (313)[Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = e. (314)By Lemma 3.2, π ( X Gal ) is not trivial.In this case, the number of branch points is 7, the number of cusps is 15, the number of nodes is20 and deg S = 10. So the first Chern number is c = 4 · c = 4 · χ = − · B.13 X degenerates to U , Figure 33:
The arrangement of planes U , . Theorem B.13. If X degenerates to U , , then π ( X Gal ) is trivial.Proof. The branch curve S in CP is an arrangement of five lines. We regenerate each vertex inturn and compute the group G .Vertex 3 is a 1-point that gives rise to braid Z ′ and to the following relation in G :Γ = Γ ′ . (315) ertex 4 is a 1-point that gives rise to braid Z ′ and to the following relation in G :Γ = Γ ′ . (316)Vertex 5 is a 1-point that gives rise to braid Z ′ and to the following relation in G :Γ = Γ ′ . (317)Vertex 6 is a 1-point that gives rise to braid Z ′ and to the following relation in G :Γ = Γ ′ . (318)Vertex 7 is a 1-point that gives rise to braid Z ′ and to the following relation in G :Γ = Γ ′ . (319)Vertex 1 is an outer 5-point, the braid monodromy corresponding to it is: e ∆ = Z
34 4 ′ , · ( Z
21 1 ′ , ) Z −
22 2 ′ , Z −
23 3 ′ , · ( Z
22 2 ′ , ) Z −
23 3 ′ , · Z
23 3 ′ , · ¯ Z
21 1 ′ , ′ · ¯ Z
22 2 ′ , ′ · ¯ Z
23 3 ′ , ′ · ( Z ′ ) Z
24 4 ′ , · ( Z
33 3 ′ , ) Z
24 4 ′ , · ( Z
21 1 ′ , ) Z −
22 2 ′ , Z
24 4 ′ , · ( Z
22 2 ′ , ) Z
24 4 ′ , · ( ¯ Z
21 1 ′ , ′ ) Z
24 4 ′ , · ( ¯ Z
22 2 ′ , ′ ) Z
24 4 ′ , · ( Z ′ ) Z
23 3 ′ , Z
24 4 ′ , · Z ′ , ′ · ( Z ′ ) Z ′ , ′ · ( Z
32 2 ′ , ) Z ′ , ′ Z
23 3 ′ , Z
24 4 ′ , · ( Z ′ ) Z
22 2 ′ , Z ′ , ′ Z
23 3 ′ , Z
24 4 ′ , · ( Z
21 1 ′ , ′ ) Z
23 3 ′ , Z
24 4 ′ , . It gives rise to the following relations: h Γ , Γ i = h Γ ′ , Γ i = h Γ − Γ ′ Γ , Γ i = e (320)[Γ ′ Γ Γ ′ Γ Γ Γ − Γ ′− Γ − Γ ′− , Γ ] = [Γ ′ Γ Γ ′ Γ Γ ′ Γ − Γ ′− Γ − Γ ′− , Γ ] = e (321)[Γ ′ Γ Γ Γ − Γ ′− , Γ ] = [Γ ′ Γ Γ ′ Γ − Γ ′− , Γ ] = e (322)[Γ , Γ ] = [Γ ′ , Γ ] = e (323)[Γ ′ Γ Γ ′ Γ Γ ′ Γ Γ Γ − Γ ′− Γ − Γ ′− Γ − Γ ′− , Γ − Γ ′ Γ ] =[Γ ′ Γ Γ ′ Γ Γ ′ Γ Γ ′ Γ − Γ ′− Γ − Γ ′− Γ − Γ ′− , Γ − Γ ′ Γ ] = e (324)[Γ ′ Γ Γ ′ Γ Γ Γ − Γ ′− Γ − Γ ′− , Γ − Γ ′ Γ ] = [Γ ′ Γ Γ ′ Γ Γ ′ Γ − Γ ′− Γ − Γ ′− , Γ − Γ ′ Γ ] = e (325)[Γ ′ Γ Γ Γ − Γ ′− , Γ − Γ ′ Γ ] = [Γ ′ Γ Γ ′ Γ − Γ ′− , Γ − Γ ′ Γ ] = e (326)Γ Γ ′ Γ Γ Γ − Γ ′− Γ − = Γ ′ (327) h Γ , Γ Γ Γ − i = h Γ ′ , Γ Γ Γ − i = h Γ − Γ ′ Γ , Γ Γ Γ − i = e (328)[Γ ′ Γ Γ Γ − Γ ′− , Γ Γ Γ − ] = [Γ ′ Γ Γ ′ Γ − Γ ′− , Γ Γ Γ − ] = e (329)[Γ , Γ Γ Γ − ] = [Γ ′ , Γ Γ Γ − ] = e (330) Γ ′ Γ Γ ′ Γ Γ Γ − Γ ′− Γ − Γ ′− , Γ Γ − Γ ′ Γ Γ − ] = [Γ ′ Γ Γ ′ Γ Γ ′ Γ − Γ ′− Γ − Γ ′− , Γ Γ − Γ ′ Γ Γ − ] = e (331)[Γ ′ Γ Γ Γ − Γ ′− , Γ Γ − Γ ′ Γ Γ − ] = [Γ ′ Γ Γ ′ Γ − Γ ′− , Γ Γ − Γ ′ Γ Γ − ] = e (332)Γ − Γ ′− Γ Γ − Γ ′ Γ Γ − Γ ′ Γ = Γ Γ Γ − (333) h Γ ′ , Γ i = h Γ ′ , Γ ′ i = h Γ ′ , Γ − Γ ′ Γ i = e (334)Γ = Γ ′ Γ Γ ′ Γ − Γ ′− (335) h Γ ′ Γ Γ ′ Γ Γ ′− Γ − Γ ′− , Γ Γ Γ − Γ Γ Γ − Γ − i = h Γ ′ Γ Γ ′ Γ ′ Γ ′− Γ − Γ ′− , Γ Γ Γ − Γ Γ Γ − Γ − i = h Γ ′ Γ Γ ′ Γ − Γ ′ Γ Γ ′− Γ − Γ ′− , Γ Γ Γ − Γ Γ Γ − Γ − i = e (336)Γ ′ Γ Γ ′ Γ − Γ ′− Γ ′− Γ − Γ ′− Γ Γ Γ − Γ − Γ ′ Γ Γ Γ − Γ − Γ ′ Γ Γ ′ Γ ′ Γ Γ ′− Γ − Γ ′− =Γ Γ Γ − Γ Γ Γ − Γ − (337)[Γ , Γ Γ Γ − Γ Γ Γ − Γ − ] = [Γ ′ , Γ Γ Γ − Γ Γ Γ − Γ − ] =[Γ , Γ Γ Γ − Γ ′ Γ Γ − Γ − ] = [Γ ′ , Γ Γ Γ − Γ ′ Γ Γ − Γ − ] = e. (338)We also have the following projective relation:Γ ′ Γ Γ ′ Γ Γ ′ Γ Γ ′ Γ Γ ′ Γ = e. (339)By (315)-(319), e G is generated by { Γ i | i = 1 , . . . , } with the following relations:Γ = Γ = Γ = Γ = Γ = e (340) h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = e (341)[Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = e. (342)Thus, e G ∼ = S , therefore π ( X Gal ) is trivial.In this case, the number of branch points is 9, the number of cusps is 12, the number of nodes is24 and deg S = 10. So the first Chern number is c = 4 · c = · χ = − · .14 X degenerates to U ,
12 345 674 3 25 61
Figure 34:
The arrangement of planes U , . Theorem B.14. If X degenerates to U , , then π ( X Gal ) is trivial.Proof. The branch curve S in CP is an arrangement of six lines. We regenerate each vertex in turnand compute the group G .Vertex 6 is a 1-point that gives rise to braid Z ′ and to the following relation in G :Γ = Γ ′ . (343)Vertex 2 is a 2-point. The braid monodromy corresponding to this point is: f ∆ = ( Z ′ ) Z
24 4 ′ , · Z
34 4 ′ , . f ∆ gives rise to the following relations: h Γ , Γ i = h Γ ′ , Γ i = h Γ − Γ ′ Γ , Γ i = e (344)Γ ′ = Γ Γ ′ Γ Γ Γ − Γ ′− Γ − . (345)Vertex 3 is a 2-point. The braid monodromy corresponding to this point is: f ∆ = ( Z ′ ) Z
22 2 ′ , · Z
32 2 ′ , . f ∆ gives rise to the following relations: h Γ , Γ i = h Γ ′ , Γ i = h Γ − Γ ′ Γ , Γ i = e (346)Γ ′ = Γ Γ ′ Γ Γ Γ − Γ ′− Γ − . (347)Vertex 4 is an inner 3-point. Its braid monodromy is f ∆ = Z ′ , ′ · Z , ′ · ( Z ′ ) Z
25 5 ′ Z ′ , ′ · ( Z ′ ) Z
25 5 ′ Z
26 6 ′ Z ′ , ′ · ( Z ′ ) Z
25 5 ′ Z , ′ · ( Z ′ ) Z
25 5 ′ Z
26 6 ′ Z , ′ . These braids give rise to the following relations in G : h Γ ′ , Γ i = h Γ ′ , Γ ′ i = h Γ ′ , Γ − Γ ′ Γ i = e (348) Γ , Γ i = h Γ , Γ ′ i = h Γ , Γ − Γ ′ Γ i = e (349)Γ ′ Γ Γ ′ Γ Γ ′− Γ − Γ ′ − = Γ ′ (350)Γ ′ Γ Γ ′ Γ ′ Γ ′− Γ − Γ ′ − = Γ ′ Γ Γ ′ − (351)Γ ′ Γ Γ Γ Γ − Γ − Γ ′ − = Γ ′ (352)Γ ′ Γ Γ Γ ′ Γ − Γ − Γ ′ − = Γ ′ Γ Γ ′− . (353)Vertex 7 is a 4-point. Its braid monodromy is e ∆ = Z
33 3 ′ , · ( Z
21 1 ′ , ) Z −
22 2 ′ , · Z
22 2 ′ , · ¯ Z
21 1 ′ , ′ · ¯ Z
22 2 ′ , ′ · ( Z ′ ) Z
23 3 ′ , · Z ′ , ′ · ( Z ′ ) Z ′ , ′ · ( Z
32 2 ′ , ) Z ′ , ′ Z
23 3 ′ , · ( Z ′ ) Z
22 2 ′ , Z ′ , ′ Z
23 3 ′ , · ( Z
21 1 ′ , ′ ) Z
23 3 ′ , . It gives rise to the following relations: h Γ , Γ i = h Γ ′ , Γ i = h Γ − Γ ′ Γ , Γ i = e (354)[Γ ′ Γ Γ Γ − Γ ′− , Γ ] = [Γ ′ Γ Γ ′ Γ − Γ ′− , Γ ] = e (355)[Γ , Γ ] = [Γ ′ , Γ ] = e (356)[Γ ′ Γ Γ ′ Γ Γ Γ − Γ ′− Γ − Γ ′− , Γ − Γ ′ Γ ] = [Γ ′ Γ Γ ′ Γ Γ ′ Γ − Γ ′− Γ − Γ ′− , Γ − Γ ′ Γ ] = e (357)[Γ ′ Γ Γ Γ − Γ ′− , Γ − Γ ′ Γ ] = [Γ ′ Γ Γ ′ Γ − Γ ′− , Γ − Γ ′ Γ ] = e (358)Γ Γ ′ Γ Γ Γ − Γ ′− Γ − = Γ ′ (359) h Γ ′ , Γ i = h Γ ′ , Γ ′ i = h Γ ′ , Γ − Γ ′ Γ i = e (360)Γ = Γ ′ Γ Γ ′ Γ − Γ ′− (361) h Γ ′ Γ Γ ′ Γ Γ ′− Γ − Γ ′− , Γ Γ Γ − i = h Γ ′ Γ Γ ′ Γ ′ Γ ′− Γ − Γ ′− , Γ Γ Γ − i == h Γ ′ Γ Γ ′ Γ − Γ ′ Γ Γ ′− Γ − Γ ′− , Γ Γ Γ − i = e (362)Γ ′ Γ Γ ′ Γ − Γ ′− Γ ′− Γ − Γ ′− Γ Γ − Γ ′ Γ Γ − Γ ′ Γ Γ ′ Γ ′ Γ Γ ′− Γ − Γ ′− = Γ Γ Γ − (363)[Γ , Γ Γ Γ − ] = [Γ ′ , Γ Γ Γ − ] = [Γ , Γ Γ ′ Γ − ] = [Γ ′ , Γ Γ ′ Γ − ] = e. (364)We also have the following parasitic and projective relations:[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (365)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (366)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (367)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (368)Γ ′ Γ Γ ′ Γ Γ ′ Γ Γ ′ Γ Γ ′ Γ Γ ′ Γ = e. (369) sing (343) and (360) in (361), we get Γ = Γ ′ . By Lemma 3.3, we get Γ = Γ ′ . Then by Lemma3.4, we have Γ = Γ ′ , Γ = Γ ′ . And again by Lemma 3.3 we get Γ = Γ ′ . e G is thus generated by { Γ i | i = 1 , . . . , } with the following relations:Γ = Γ = Γ = Γ = Γ = Γ = e (370) h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = e (371)[Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = e (372)Γ = Γ Γ Γ . (373)By Remark 2.8 in [24], we get also the relations[Γ , Γ Γ Γ ] = [Γ , Γ Γ Γ ] = e. (374)By Theorem 2.3 in [24] we have e G ∼ = S , therefore π ( X Gal ) is trivial.In this case, the number of branch points is 10, the number of cusps is 21, the number of nodes is28 and deg S = 12. So the first Chern number is c = 9 · c = · χ = − · B.15 X degenerates to U ,
12 345 671 2 34 56
Figure 35:
The arrangement of planes U , . Theorem B.15. If X degenerates to U , , then π ( X Gal ) is trivial.Proof. The branch curve S in CP is an arrangement of six lines. We regenerate each vertex in turnand compute the group G .Vertex 2 is an outer 3-point. The braid monodromy corresponding to this 3-point is: f ∆ = Z
21 1 ′ , ′ · Z ′ , ′ · ( Z ′ ) Z ′ , ′ · ( Z
34 4 ′ , ) Z ′ , ′ · ( Z ′ ) Z
24 4 ′ , Z ′ , ′ . ∆ thus gives rise to the following relations: h Γ ′ , Γ i = h Γ ′ , Γ ′ i = h Γ ′ , Γ − Γ ′ Γ i = e (375)Γ = Γ ′ Γ Γ ′ Γ − Γ ′ − (376) h Γ , Γ ′ Γ Γ ′ Γ Γ ′ − Γ − Γ ′ − i = h Γ , Γ ′ Γ Γ ′ Γ ′ Γ ′ − Γ − Γ ′ − i == h Γ , Γ ′ Γ Γ ′ Γ − Γ ′ Γ Γ ′ − Γ − Γ ′ − i = e (377)Γ ′ = Γ Γ ′ Γ Γ ′ Γ ′ Γ Γ ′ − Γ − Γ ′ − Γ Γ ′ Γ Γ ′ Γ − Γ ′ − Γ ′ − Γ − Γ ′ − Γ − (378)[Γ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ] = [Γ ′ , Γ ′ ] = e. (379)Vertex 3 is an outer 3-point. The braid monodromy corresponding to this 3-point is: f ∆ = Z
23 3 ′ , ′ · Z ′ , ′ · ( Z ′ ) Z ′ , ′ · ( Z
35 5 ′ , ) Z ′ , ′ · ( Z ′ ) Z
25 5 ′ , Z ′ , ′ . f ∆ thus gives rise to the following relations: h Γ ′ , Γ i = h Γ ′ , Γ ′ i = h Γ ′ , Γ − Γ ′ Γ i = e (380)Γ = Γ ′ Γ Γ ′ Γ − Γ ′ − (381) h Γ , Γ ′ Γ Γ ′ Γ Γ ′ − Γ − Γ ′ − i = h Γ , Γ ′ Γ Γ ′ Γ ′ Γ ′ − Γ − Γ ′ − i == h Γ , Γ ′ Γ Γ ′ Γ − Γ ′ Γ Γ ′ − Γ − Γ ′ − i = e (382)Γ ′ = Γ Γ ′ Γ Γ ′ Γ ′ Γ Γ ′ − Γ − Γ ′ − Γ Γ ′ Γ Γ ′ Γ − Γ ′ − Γ ′ − Γ − Γ ′ − Γ − (383)[Γ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ] = [Γ ′ , Γ ′ ] = e. (384)Vertex 4 is an inner 3-point. Its braid monodromy is f ∆ = Z ′ , ′ · Z , ′ · ( Z ′ ) Z
24 4 ′ Z ′ , ′ · ( Z ′ ) Z
24 4 ′ Z
25 5 ′ Z ′ , ′ · ( Z ′ ) Z
24 4 ′ Z , ′ · ( Z ′ ) Z
24 4 ′ Z
25 5 ′ Z , ′ . These braids give rise to the following relations in G : h Γ ′ , Γ i = h Γ ′ , Γ ′ i = h Γ ′ , Γ − Γ ′ Γ i = e (385) h Γ , Γ i = h Γ , Γ ′ i = h Γ , Γ − Γ ′ Γ i = e (386)Γ ′ Γ Γ ′ Γ Γ ′− Γ − Γ ′ − = Γ ′ (387)Γ ′ Γ Γ ′ Γ ′ Γ ′− Γ − Γ ′ − = Γ ′ Γ Γ ′ − (388)Γ ′ Γ Γ Γ Γ − Γ − Γ ′ − = Γ ′ (389)Γ ′ Γ Γ Γ ′ Γ − Γ − Γ ′ − = Γ ′ Γ Γ ′− . (390)Vertex 7 is an outer 3-point. The braid monodromy corresponding to this 3-point is: f ∆ = Z
21 1 ′ , ′ · Z ′ , ′ · ( Z ′ ) Z ′ , ′ · ( Z
32 2 ′ , ) Z ′ , ′ · ( Z ′ ) Z
22 2 ′ , Z ′ , ′ . ∆ thus gives rise to the following relations: h Γ ′ , Γ i = h Γ ′ , Γ ′ i = h Γ ′ , Γ − Γ ′ Γ i = e (391)Γ = Γ ′ Γ Γ ′ Γ − Γ ′ − (392) h Γ , Γ ′ Γ Γ ′ Γ Γ ′ − Γ − Γ ′ − i = h Γ , Γ ′ Γ Γ ′ Γ ′ Γ ′ − Γ − Γ ′ − i == h Γ , Γ ′ Γ Γ ′ Γ − Γ ′ Γ Γ ′ − Γ − Γ ′ − i = e (393)Γ ′ = Γ Γ ′ Γ Γ ′ Γ ′ Γ Γ ′ − Γ − Γ ′ − Γ Γ ′ Γ Γ ′ Γ − Γ ′ − Γ ′ − Γ − Γ ′ − Γ − (394)[Γ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ] = [Γ ′ , Γ ′ ] = e. (395)We also have the following parasitic and projective relations:[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (396)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (397)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (398)Γ ′ Γ Γ ′ Γ Γ ′ Γ Γ ′ Γ Γ ′ Γ Γ ′ Γ = e. (399)Equating (387) and (389) and using (385) and (386), we get Γ = Γ ′ . By Lemma 3.5, we getΓ = Γ ′ , Γ = Γ ′ , Γ = Γ ′ , Γ = Γ ′ and Γ = Γ ′ . e G is thus generated by { Γ i | i = 1 , . . . , } with the following relations:Γ = Γ = Γ = Γ = Γ = Γ = e (400) h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = h Γ , Γ i == h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = e (401)[Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = e (402)Γ = Γ Γ Γ . (403)By Remark 2.8 in [24], we get also the relations[Γ , Γ Γ Γ ] = [Γ , Γ Γ Γ ] = [Γ , Γ Γ Γ ] = e. (404)By Theorem 2.3 in [24] we have e G ∼ = S , therefore π ( X Gal ) is trivial.In this case, the number of branch points is 10, the number of cusps is 24, the number of nodes is24 and deg S = 12. So the first Chern number is c = 9 · c = 6 · χ = − .16 X degenerates to U ,
12 345 676 3 24 5 1
Figure 36:
The arrangement of planes U , . Theorem B.16. If X degenerates to U , , then π ( X Gal ) is trivial.Proof. The branch curve S in CP is an arrangement of six lines. We regenerate each vertex in turnand compute the group G .Vertex 6 is a 1-point that gives rise to braid Z ′ and to the following relation in G :Γ = Γ ′ . (405)Vertex 2 is a 2-point. The braid monodromy corresponding to this point is: f ∆ = ( Z ′ ) Z
24 4 ′ , · Z
34 4 ′ , . f ∆ gives rise to the following relations: h Γ , Γ i = h Γ ′ , Γ i = h Γ − Γ ′ Γ , Γ i = e (406)Γ ′ = Γ Γ ′ Γ Γ Γ − Γ ′− Γ − . (407)Vertex 3 is an outer 3-point. The braid monodromy corresponding to this 3-point is: f ∆ = Z
21 1 ′ , ′ · Z ′ , ′ · ( Z ′ ) Z ′ , ′ · ( Z
32 2 ′ , ) Z ′ , ′ · ( Z ′ ) Z
22 2 ′ , Z ′ , ′ . f ∆ thus gives rise to the following relations: h Γ ′ , Γ i = h Γ ′ , Γ ′ i = h Γ ′ , Γ − Γ ′ Γ i = e (408)Γ = Γ ′ Γ Γ ′ Γ − Γ ′ − (409) h Γ , Γ ′ Γ Γ ′ Γ Γ ′ − Γ − Γ ′ − i = h Γ , Γ ′ Γ Γ ′ Γ ′ Γ ′ − Γ − Γ ′ − i == h Γ , Γ ′ Γ Γ ′ Γ − Γ ′ Γ Γ ′ − Γ − Γ ′ − i = e (410)Γ ′ = Γ Γ ′ Γ Γ ′ Γ ′ Γ Γ ′ − Γ − Γ ′ − Γ Γ ′ Γ Γ ′ Γ − Γ ′ − Γ ′ − Γ − Γ ′ − Γ − (411)[Γ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ] = [Γ ′ , Γ ′ ] = e. (412) ertex 4 is an inner 3-point. Its braid monodromy is f ∆ = Z ′ , ′ · Z , ′ · ( Z ′ ) Z
24 4 ′ Z ′ , ′ · ( Z ′ ) Z
24 4 ′ Z
25 5 ′ Z ′ , ′ · ( Z ′ ) Z
24 4 ′ Z , ′ · ( Z ′ ) Z
24 4 ′ Z
25 5 ′ Z , ′ . These braids give rise to the following relations in G : h Γ ′ , Γ i = h Γ ′ , Γ ′ i = h Γ ′ , Γ − Γ ′ Γ i = e (413) h Γ , Γ i = h Γ , Γ ′ i = h Γ , Γ − Γ ′ Γ i = e (414)Γ ′ Γ Γ ′ Γ Γ ′− Γ − Γ ′ − = Γ ′ (415)Γ ′ Γ Γ ′ Γ ′ Γ ′− Γ − Γ ′ − = Γ ′ Γ Γ ′ − (416)Γ ′ Γ Γ Γ Γ − Γ − Γ ′ − = Γ ′ (417)Γ ′ Γ Γ Γ ′ Γ − Γ − Γ ′ − = Γ ′ Γ Γ ′− . (418)Vertex 7 is an outer 3-point. The braid monodromy corresponding to this 3-point is: f ∆ = Z
22 2 ′ , ′ · Z ′ , ′ · ( Z ′ ) Z ′ , ′ · ( Z
33 3 ′ , ) Z ′ , ′ · ( Z ′ ) Z
23 3 ′ , Z ′ , ′ . f ∆ thus gives rise to the following relations: h Γ ′ , Γ i = h Γ ′ , Γ ′ i = h Γ ′ , Γ − Γ ′ Γ i = e (419)Γ = Γ ′ Γ Γ ′ Γ − Γ ′ − (420) h Γ , Γ ′ Γ Γ ′ Γ Γ ′ − Γ − Γ ′ − i = h Γ , Γ ′ Γ Γ ′ Γ ′ Γ ′ − Γ − Γ ′ − i == h Γ , Γ ′ Γ Γ ′ Γ − Γ ′ Γ Γ ′ − Γ − Γ ′ − i = e (421)Γ ′ = Γ Γ ′ Γ Γ ′ Γ ′ Γ Γ ′ − Γ − Γ ′ − Γ Γ ′ Γ Γ ′ Γ − Γ ′ − Γ ′ − Γ − Γ ′ − Γ − (422)[Γ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ] = [Γ ′ , Γ ′ ] = e. (423)We also have the following parasitic and projective relations:[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (424)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (425)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (426)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (427)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (428)Γ ′ Γ Γ ′ Γ Γ ′ Γ Γ ′ Γ Γ ′ Γ Γ ′ Γ = e. (429)By Lemma 3.5, we get Γ = Γ ′ , Γ = Γ ′ and Γ = Γ ′ , Γ = Γ ′ . Then by Lemma 3.3, we getΓ = Γ ′ . G is thus generated by { Γ i | i = 1 , . . . , } with the following relations:Γ = Γ = Γ = Γ = Γ = Γ = e (430) h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = e (431)[Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = e (432)Γ = Γ Γ Γ . (433)By Remark 2.8 in [24], we get also the relations[Γ , Γ Γ Γ ] = e. (434)By Theorem 2.3 in [24] we have e G ∼ = S , therefore π ( X Gal ) is trivial.In this case, the number of branch points is 10, the number of cusps is 21, the number of nodes is28 and deg S = 12. So the first Chern number is c = 9 · c = · χ = − · B.17 X degenerates to U ,
12 34 5 676 5 43 21
Figure 37:
The arrangement of planes U , . Theorem B.17. If X degenerates to U , , then π ( X Gal ) is trivial.Proof. The branch curve S in CP is an arrangement of six lines. We regenerate each vertex in turnand compute the group G .Vertex 2 is a 1-point that gives rise to braid Z ′ and to the following relation in G :Γ = Γ ′ . (435)Vertex 5 is a 1-point that gives rise to braid Z ′ and to the following relation in G :Γ = Γ ′ . (436) ertex 6 is a 1-point that gives rise to braid Z ′ and to the following relation in G :Γ = Γ ′ . (437)Vertex 1 is a 2-point. The braid monodromy corresponding to this point is: f ∆ = ( Z ′ ) Z
23 3 ′ , · Z
33 3 ′ , . f ∆ gives rise to the following relations: h Γ , Γ i = h Γ ′ , Γ i = h Γ − Γ ′ Γ , Γ i = e (438)Γ ′ = Γ Γ ′ Γ Γ Γ − Γ ′− Γ − . (439)Vertex 4 is an inner 3-point. Its braid monodromy is f ∆ = Z ′ , ′ · Z , ′ · ( Z ′ ) Z
25 5 ′ Z ′ , ′ · ( Z ′ ) Z
25 5 ′ Z
26 6 ′ Z ′ , ′ · ( Z ′ ) Z
25 5 ′ Z , ′ · ( Z ′ ) Z
25 5 ′ Z
26 6 ′ Z , ′ . These braids give rise to the following relations in G : h Γ ′ , Γ i = h Γ ′ , Γ ′ i = h Γ ′ , Γ − Γ ′ Γ i = e (440) h Γ , Γ i = h Γ , Γ ′ i = h Γ , Γ − Γ ′ Γ i = e (441)Γ ′ Γ Γ ′ Γ Γ ′− Γ − Γ ′ − = Γ ′ (442)Γ ′ Γ Γ ′ Γ ′ Γ ′− Γ − Γ ′ − = Γ ′ Γ Γ ′ − (443)Γ ′ Γ Γ Γ Γ − Γ − Γ ′ − = Γ ′ (444)Γ ′ Γ Γ Γ ′ Γ − Γ − Γ ′ − = Γ ′ Γ Γ ′− . (445)Vertex 3 is a 4-point. Its braid monodromy is e ∆ = Z
33 3 ′ , · ( Z
21 1 ′ , ) Z −
22 2 ′ , · Z
22 2 ′ , · ¯ Z
21 1 ′ , ′ · ¯ Z
22 2 ′ , ′ · ( Z ′ ) Z
23 3 ′ , · Z ′ , ′ · ( Z ′ ) Z ′ , ′ · ( Z
32 2 ′ , ) Z ′ , ′ Z
23 3 ′ , · ( Z ′ ) Z
22 2 ′ , Z ′ , ′ Z
23 3 ′ , · ( Z
21 1 ′ , ′ ) Z
23 3 ′ , . It gives rise to the following relations: h Γ , Γ i = h Γ ′ , Γ i = h Γ − Γ ′ Γ , Γ i = e (446)[Γ ′ Γ Γ Γ − Γ ′− , Γ ] = [Γ ′ Γ Γ ′ Γ − Γ ′− , Γ ] = e (447)[Γ , Γ ] = [Γ ′ , Γ ] = e (448)[Γ ′ Γ Γ ′ Γ Γ Γ − Γ ′− Γ − Γ ′− , Γ − Γ ′ Γ ] = [Γ ′ Γ Γ ′ Γ Γ ′ Γ − Γ ′− Γ − Γ ′− , Γ − Γ ′ Γ ] = e (449)[Γ ′ Γ Γ Γ − Γ ′− , Γ − Γ ′ Γ ] = [Γ ′ Γ Γ ′ Γ − Γ ′− , Γ − Γ ′ Γ ] = e (450) Γ ′ Γ Γ Γ − Γ ′− Γ − = Γ ′ (451) h Γ ′ , Γ i = h Γ ′ , Γ ′ i = h Γ ′ , Γ − Γ ′ Γ i = e (452)Γ = Γ ′ Γ Γ ′ Γ − Γ ′− (453) h Γ ′ Γ Γ ′ Γ Γ ′− Γ − Γ ′− , Γ Γ Γ − i = h Γ ′ Γ Γ ′ Γ ′ Γ ′− Γ − Γ ′− , Γ Γ Γ − i == h Γ ′ Γ Γ ′ Γ − Γ ′ Γ Γ ′− Γ − Γ ′− , Γ Γ Γ − i = e (454)Γ ′ Γ Γ ′ Γ − Γ ′− Γ ′− Γ − Γ ′− Γ Γ − Γ ′ Γ Γ − Γ ′ Γ Γ ′ Γ ′ Γ Γ ′− Γ − Γ ′− = Γ Γ Γ − (455)[Γ , Γ Γ Γ − ] = [Γ ′ , Γ Γ Γ − ] = [Γ , Γ Γ ′ Γ − ] = [Γ ′ , Γ Γ ′ Γ − ] = e. (456)We also have the following parasitic and projective relations:[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (457)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (458)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (459)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (460)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (461)Γ ′ Γ Γ ′ Γ Γ ′ Γ Γ ′ Γ Γ ′ Γ Γ ′ Γ = e. (462)By Lemma 3.4, we get Γ = Γ ′ , Γ = Γ ′ . Then by Lemma 3.3, we have Γ = Γ ′ . e G is thus generated by { Γ i | i = 1 , . . . , } with the following relations:Γ = Γ = Γ = Γ = Γ = Γ = e (463) h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = e (464)[Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = e (465)Γ = Γ Γ Γ . (466)By Remark 2.8 in [24], we get also the relation[Γ , Γ Γ Γ ] = e. (467)By Theorem 2.3 in [24] we have e G ∼ = S , therefore π ( X Gal ) is trivial.In this case, the number of branch points is 11, the number of cusps is 18, the number of nodes is32 and deg S = 12. So the first Chern number is c = 9 · c = · χ = − · .18 X degenerates to U ,
12 34 5 672 1 34 56
Figure 38:
The arrangement of planes U , . Theorem B.18. If X degenerates to U , , then π ( X Gal ) is not trivial.Proof. The branch curve S in CP is an arrangement of six lines. We regenerate each vertex in turnand compute the group G .Vertex 2 is a 1-point that gives rise to braid Z ′ and to the following relation in G :Γ = Γ ′ . (468)Vertex 5 is a 2-point. The braid monodromy corresponding to this point is: f ∆ = ( Z ′ ) Z
25 5 ′ , · Z
35 5 ′ , . f ∆ gives rise to the following relations: h Γ , Γ i = h Γ ′ , Γ i = h Γ − Γ ′ Γ , Γ i = e (469)Γ ′ = Γ Γ ′ Γ Γ Γ − Γ ′− Γ − . (470)Vertex 1 is an outer 3-point. The braid monodromy corresponding to this 3-point is: f ∆ = Z
21 1 ′ , ′ · Z ′ , ′ · ( Z ′ ) Z ′ , ′ · ( Z
34 4 ′ , ) Z ′ , ′ · ( Z ′ ) Z
24 4 ′ , Z ′ , ′ . f ∆ thus gives rise to the following relations: h Γ ′ , Γ i = h Γ ′ , Γ ′ i = h Γ ′ , Γ − Γ ′ Γ i = e (471)Γ = Γ ′ Γ Γ ′ Γ − Γ ′ − (472) h Γ , Γ ′ Γ Γ ′ Γ Γ ′ − Γ − Γ ′ − i = h Γ , Γ ′ Γ Γ ′ Γ ′ Γ ′ − Γ − Γ ′ − i == h Γ , Γ ′ Γ Γ ′ Γ − Γ ′ Γ Γ ′ − Γ − Γ ′ − i = e (473)Γ ′ = Γ Γ ′ Γ Γ ′ Γ ′ Γ Γ ′ − Γ − Γ ′ − Γ Γ ′ Γ Γ ′ Γ − Γ ′ − Γ ′ − Γ − Γ ′ − Γ − (474)[Γ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ] = [Γ ′ , Γ ′ ] = e. (475) ertex 3 is an outer 3-point. The braid monodromy corresponding to this 3-point is: f ∆ = Z
23 3 ′ , ′ · Z ′ , ′ · ( Z ′ ) Z ′ , ′ · ( Z
34 4 ′ , ) Z ′ , ′ · ( Z ′ ) Z
24 4 ′ , Z ′ , ′ . f ∆ thus gives rise to the following relations: h Γ ′ , Γ i = h Γ ′ , Γ ′ i = h Γ ′ , Γ − Γ ′ Γ i = e (476)Γ = Γ ′ Γ Γ ′ Γ − Γ ′ − (477) h Γ , Γ ′ Γ Γ ′ Γ Γ ′ − Γ − Γ ′ − i = h Γ , Γ ′ Γ Γ ′ Γ ′ Γ ′ − Γ − Γ ′ − i == h Γ , Γ ′ Γ Γ ′ Γ − Γ ′ Γ Γ ′ − Γ − Γ ′ − i = e (478)Γ ′ = Γ Γ ′ Γ Γ ′ Γ ′ Γ Γ ′ − Γ − Γ ′ − Γ Γ ′ Γ Γ ′ Γ − Γ ′ − Γ ′ − Γ − Γ ′ − Γ − (479)[Γ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ] = [Γ ′ , Γ ′ ] = e. (480)Vertex 4 is an inner 3-point. Its braid monodromy is f ∆ = Z ′ , ′ · Z , ′ · ( Z ′ ) Z
22 2 ′ Z ′ , ′ · ( Z ′ ) Z
22 2 ′ Z
23 3 ′ Z ′ , ′ · ( Z ′ ) Z
22 2 ′ Z , ′ · ( Z ′ ) Z
22 2 ′ Z
23 3 ′ Z , ′ . These braids give rise to the following relations in G : h Γ ′ , Γ i = h Γ ′ , Γ ′ i = h Γ ′ , Γ − Γ ′ Γ i = e (481) h Γ , Γ i = h Γ , Γ ′ i = h Γ , Γ − Γ ′ Γ i = e (482)Γ ′ Γ Γ ′ Γ Γ ′− Γ − Γ ′ − = Γ ′ (483)Γ ′ Γ Γ ′ Γ ′ Γ ′− Γ − Γ ′ − = Γ ′ Γ Γ ′ − (484)Γ ′ Γ Γ Γ Γ − Γ − Γ ′ − = Γ ′ (485)Γ ′ Γ Γ Γ ′ Γ − Γ − Γ ′ − = Γ ′ Γ Γ ′− . (486)We also have the following parasitic and projective relations:[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (487)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (488)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (489)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (490)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (491)Γ ′ Γ Γ ′ Γ Γ ′ Γ Γ ′ Γ Γ ′ Γ Γ ′ Γ = e. (492)By Lemma 3.4, we get Γ = Γ ′ , Γ = Γ ′ . Then by Lemma 3.5, we get Γ = Γ ′ , Γ = Γ ′ andΓ = Γ ′ . G is thus generated by { Γ i | i = 1 , . . . , } with the following relations:Γ = Γ = Γ = Γ = Γ = Γ = e (493) h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = e (494)[Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = e (495)Γ = Γ Γ Γ . (496)By Lemma 3.2, π ( X Gal ) is not trivial.In this case, the number of branch points is 10, the number of cusps is 21, the number of nodes is28 and deg S = 12. So the first Chern number is c = 9 · c = · χ = − · B.19 X degenerates to U ,
12 34 5 672 1 34 5 6
Figure 39:
The arrangement of planes U , . Theorem B.19. If X degenerates to U , , then π ( X Gal ) is trivial.Proof. The branch curve S in CP is an arrangement of six lines. We regenerate each vertex in turnand compute the group G .Vertex 2 is a 1-point that gives rise to braid Z ′ and to the following relation in G :Γ = Γ ′ . (497)Vertex 6 is a 1-point that gives rise to braid Z ′ and to the following relation in G :Γ = Γ ′ . (498)Vertex 1 is a 2-point. The braid monodromy corresponding to this point is: f ∆ = ( Z ′ ) Z
21 1 ′ , · Z
31 1 ′ , . ∆ gives rise to the following relations: h Γ , Γ i = h Γ ′ , Γ i = h Γ − Γ ′ Γ , Γ i = e (499)Γ ′ = Γ Γ ′ Γ Γ Γ − Γ ′− Γ − . (500)Vertex 5 is a 2-point. The braid monodromy corresponding to this point is: f ∆ = ( Z ′ ) Z
25 5 ′ , · Z
35 5 ′ , . f ∆ gives rise to the following relations: h Γ , Γ i = h Γ ′ , Γ i = h Γ − Γ ′ Γ , Γ i = e (501)Γ ′ = Γ Γ ′ Γ Γ Γ − Γ ′− Γ − . (502)Vertex 3 is an outer 3-point. The braid monodromy corresponding to this 3-point is: f ∆ = Z
23 3 ′ , ′ · Z ′ , ′ · ( Z ′ ) Z ′ , ′ · ( Z
34 4 ′ , ) Z ′ , ′ · ( Z ′ ) Z
24 4 ′ , Z ′ , ′ . f ∆ thus gives rise to the following relations: h Γ ′ , Γ i = h Γ ′ , Γ ′ i = h Γ ′ , Γ − Γ ′ Γ i = e (503)Γ = Γ ′ Γ Γ ′ Γ − Γ ′ − (504) h Γ , Γ ′ Γ Γ ′ Γ Γ ′ − Γ − Γ ′ − i = h Γ , Γ ′ Γ Γ ′ Γ ′ Γ ′ − Γ − Γ ′ − i == h Γ , Γ ′ Γ Γ ′ Γ − Γ ′ Γ Γ ′ − Γ − Γ ′ − i = e (505)Γ ′ = Γ Γ ′ Γ Γ ′ Γ ′ Γ Γ ′ − Γ − Γ ′ − Γ Γ ′ Γ Γ ′ Γ − Γ ′ − Γ ′ − Γ − Γ ′ − Γ − (506)[Γ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ] = [Γ ′ , Γ ′ ] = e. (507)Vertex 4 is an inner 3-point. Its braid monodromy is f ∆ = Z ′ , ′ · Z , ′ · ( Z ′ ) Z
22 2 ′ Z ′ , ′ · ( Z ′ ) Z
22 2 ′ Z
23 3 ′ Z ′ , ′ · ( Z ′ ) Z
22 2 ′ Z , ′ · ( Z ′ ) Z
22 2 ′ Z
23 3 ′ Z , ′ . These braids give rise to the following relations in G : h Γ ′ , Γ i = h Γ ′ , Γ ′ i = h Γ ′ , Γ − Γ ′ Γ i = e (508) h Γ , Γ i = h Γ , Γ ′ i = h Γ , Γ − Γ ′ Γ i = e (509)Γ ′ Γ Γ ′ Γ Γ ′− Γ − Γ ′ − = Γ ′ (510)Γ ′ Γ Γ ′ Γ ′ Γ ′− Γ − Γ ′ − = Γ ′ Γ Γ ′ − (511)Γ ′ Γ Γ Γ Γ − Γ − Γ ′ − = Γ ′ (512)Γ ′ Γ Γ Γ ′ Γ − Γ − Γ ′ − = Γ ′ Γ Γ ′− . (513) e also have the following parasitic and projective relations:[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (514)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (515)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (516)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (517)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (518)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (519)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (520)Γ ′ Γ Γ ′ Γ Γ ′ Γ Γ ′ Γ Γ ′ Γ Γ ′ Γ = e. (521)By Lemma 3.4, we get Γ = Γ ′ , Γ = Γ ′ . By Lemma 3.3, we get Γ = Γ ′ and Γ = Γ ′ . e G is thus generated by { Γ i | i = 1 , . . . , } with the following relations:Γ = Γ = Γ = Γ = Γ = Γ = e (522) h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = e (523)[Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = e (524)Γ = Γ Γ Γ . (525)By Remark 2.8 in [24], we get also the relation[Γ , Γ Γ Γ ] = e. (526)By Theorem 2.3 in [24] we have e G ∼ = S , therefore π ( X Gal ) is trivial.In this case, the number of branch points is 10, the number of cusps is 18, the number of nodes is32 and deg S = 12. So the first Chern number is c = 9 · c = 7 · χ = − · .20 X degenerates to U ∪ Figure 40:
The arrangement of planes U ∪ . Theorem B.20. If X degenerates to U ∪ , then π ( X Gal ) is trivial.Proof. The branch curve S in CP is an arrangement of seven lines. We regenerate each vertex inturn and compute the group G .Vertex 1 is a 1-point that gives rise to braid Z ′ and to the following relation in G :Γ = Γ ′ . (527)Vertex 6 is a 1-point that gives rise to braid Z ′ and to the following relation in G :Γ = Γ ′ . (528)Vertex 2 is an inner 3-point. Its braid monodromy is f ∆ = Z ′ , ′ · Z , ′ · ( Z ′ ) Z
22 2 ′ Z ′ , ′ · ( Z ′ ) Z
22 2 ′ Z
23 3 ′ Z ′ , ′ · ( Z ′ ) Z
22 2 ′ Z , ′ · ( Z ′ ) Z
22 2 ′ Z
23 3 ′ Z , ′ . These braids give rise to the following relations in G : h Γ ′ , Γ i = h Γ ′ , Γ ′ i = h Γ ′ , Γ − Γ ′ Γ i = e (529) h Γ , Γ i = h Γ , Γ ′ i = h Γ , Γ − Γ ′ Γ i = e (530)Γ ′ Γ Γ ′ Γ Γ ′− Γ − Γ ′ − = Γ ′ (531)Γ ′ Γ Γ ′ Γ ′ Γ ′− Γ − Γ ′ − = Γ ′ Γ Γ ′ − (532)Γ ′ Γ Γ Γ Γ − Γ − Γ ′ − = Γ ′ (533)Γ ′ Γ Γ Γ ′ Γ − Γ − Γ ′ − = Γ ′ Γ Γ ′− . (534)Vertex 3 is an outer 3-point. The braid monodromy corresponding to this 3-point is: f ∆ = Z
21 1 ′ , ′ · Z ′ , ′ · ( Z ′ ) Z ′ , ′ · ( Z
34 4 ′ , ) Z ′ , ′ · ( Z ′ ) Z
24 4 ′ , Z ′ , ′ . ∆ thus gives rise to the following relations: h Γ ′ , Γ i = h Γ ′ , Γ ′ i = h Γ ′ , Γ − Γ ′ Γ i = e (535)Γ = Γ ′ Γ Γ ′ Γ − Γ ′ − (536) h Γ , Γ ′ Γ Γ ′ Γ Γ ′ − Γ − Γ ′ − i = h Γ , Γ ′ Γ Γ ′ Γ ′ Γ ′ − Γ − Γ ′ − i == h Γ , Γ ′ Γ Γ ′ Γ − Γ ′ Γ Γ ′ − Γ − Γ ′ − i = e (537)Γ ′ = Γ Γ ′ Γ Γ ′ Γ ′ Γ Γ ′ − Γ − Γ ′ − Γ Γ ′ Γ Γ ′ Γ − Γ ′ − Γ ′ − Γ − Γ ′ − Γ − (538)[Γ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ] = [Γ ′ , Γ ′ ] = e. (539)Vertex 4 is an outer 3-point. The braid monodromy corresponding to this 3-point is: f ∆ = Z
23 3 ′ , ′ · Z ′ , ′ · ( Z ′ ) Z ′ , ′ · ( Z
34 4 ′ , ) Z ′ , ′ · ( Z ′ ) Z
24 4 ′ , Z ′ , ′ . f ∆ thus gives rise to the following relations: h Γ ′ , Γ i = h Γ ′ , Γ ′ i = h Γ ′ , Γ − Γ ′ Γ i = e (540)Γ = Γ ′ Γ Γ ′ Γ − Γ ′ − (541) h Γ , Γ ′ Γ Γ ′ Γ Γ ′ − Γ − Γ ′ − i = h Γ , Γ ′ Γ Γ ′ Γ ′ Γ ′ − Γ − Γ ′ − i == h Γ , Γ ′ Γ Γ ′ Γ − Γ ′ Γ Γ ′ − Γ − Γ ′ − i = e (542)Γ ′ = Γ Γ ′ Γ Γ ′ Γ ′ Γ Γ ′ − Γ − Γ ′ − Γ Γ ′ Γ Γ ′ Γ − Γ ′ − Γ ′ − Γ − Γ ′ − Γ − (543)[Γ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ] = [Γ ′ , Γ ′ ] = e. (544)Vertex 5 is an inner 3-point. Its braid monodromy is f ∆ = Z ′ , ′ · Z , ′ · ( Z ′ ) Z
26 6 ′ Z ′ , ′ · ( Z ′ ) Z
26 6 ′ Z
27 7 ′ Z ′ , ′ · ( Z ′ ) Z
26 6 ′ Z , ′ · ( Z ′ ) Z
26 6 ′ Z
27 7 ′ Z , ′ . These braids give rise to the following relations in G : h Γ ′ , Γ i = h Γ ′ , Γ ′ i = h Γ ′ , Γ − Γ ′ Γ i = e (545) h Γ , Γ i = h Γ , Γ ′ i = h Γ , Γ − Γ ′ Γ i = e (546)Γ ′ Γ Γ ′ Γ Γ ′− Γ − Γ ′ − = Γ ′ (547)Γ ′ Γ Γ ′ Γ ′ Γ ′− Γ − Γ ′ − = Γ ′ Γ Γ ′ − (548)Γ ′ Γ Γ Γ Γ − Γ − Γ ′ − = Γ ′ (549)Γ ′ Γ Γ Γ ′ Γ − Γ − Γ ′ − = Γ ′ Γ Γ ′− . (550)We also have the following parasitic and projective relations:[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (551) Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (552)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (553)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (554)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (555)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (556)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (557)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (558)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (559)Γ ′ Γ Γ ′ Γ Γ ′ Γ Γ ′ Γ Γ ′ Γ Γ ′ Γ Γ ′ Γ = e. (560)By Lemma 3.4, we get Γ = Γ ′ , Γ = Γ ′ and Γ = Γ ′ , Γ = Γ ′ . Then by Lemma 3.5, we getΓ = Γ ′ . e G is thus generated by { Γ i | i = 1 , . . . , } with the following relations:Γ = Γ = Γ = Γ = Γ = Γ = Γ = e (561) h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = h Γ , Γ i == h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = e (562)[Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] == [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = e (563)Γ = Γ Γ Γ (564)Γ = Γ Γ Γ . (565)By Remark 2.8 in [24], we get also the relations[Γ , Γ Γ Γ ] = [Γ , Γ Γ Γ ] = e. (566)By Theorem 2.3 in [24] we have e G ∼ = S , therefore π ( X Gal ) is trivial.In this case, the number of branch points is 14, the number of cusps is 24, the number of nodes is44 and deg S = 14. So the first Chern number is c = 16 · c = 11 · χ = − · .21 X degenerates to U , Figure 41:
The arrangement of planes U , . Theorem B.21. If X degenerates to U , , then π ( X Gal ) is trivial.Proof. The branch curve S in CP is an arrangement of six lines. We regenerate each vertex in turnand compute the group G .Vertex 3 is a 1-point that gives rise to braid Z ′ and to the following relation in G :Γ = Γ ′ . (567)Vertex 1 is a 2-point. The braid monodromy corresponding to this point is: f ∆ = ( Z ′ ) Z
21 1 ′ , · Z
31 1 ′ , . f ∆ gives rise to the following relations: h Γ , Γ i = h Γ ′ , Γ i = h Γ − Γ ′ Γ , Γ i = e (568)Γ ′ = Γ Γ ′ Γ Γ Γ − Γ ′− Γ − . (569)Vertex 6 is a 2-point. The braid monodromy corresponding to this point is: f ∆ = ( Z ′ ) Z
25 5 ′ , · Z
35 5 ′ , . f ∆ gives rise to the following relations: h Γ , Γ i = h Γ ′ , Γ i = h Γ − Γ ′ Γ , Γ i = e (570)Γ ′ = Γ Γ ′ Γ Γ Γ − Γ ′− Γ − . (571)Vertex 5 is an outer 3-point. The braid monodromy corresponding to this 3-point is: f ∆ = Z
21 1 ′ , ′ · Z ′ , ′ · ( Z ′ ) Z ′ , ′ · ( Z
34 4 ′ , ) Z ′ , ′ · ( Z ′ ) Z
24 4 ′ , Z ′ , ′ . ∆ thus gives rise to the following relations: h Γ ′ , Γ i = h Γ ′ , Γ ′ i = h Γ ′ , Γ − Γ ′ Γ i = e (572)Γ = Γ ′ Γ Γ ′ Γ − Γ ′ − (573) h Γ , Γ ′ Γ Γ ′ Γ Γ ′ − Γ − Γ ′ − i = h Γ , Γ ′ Γ Γ ′ Γ ′ Γ ′ − Γ − Γ ′ − i == h Γ , Γ ′ Γ Γ ′ Γ − Γ ′ Γ Γ ′ − Γ − Γ ′ − i = e (574)Γ ′ = Γ Γ ′ Γ Γ ′ Γ ′ Γ Γ ′ − Γ − Γ ′ − Γ Γ ′ Γ Γ ′ Γ − Γ ′ − Γ ′ − Γ − Γ ′ − Γ − (575)[Γ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ] = [Γ ′ , Γ ′ ] = e. (576)Vertex 4 is an inner 4-point. Its braid monodromy appears in [8]. The corresponding relations in G are: h Γ ′ , Γ i = h Γ ′ , Γ ′ i = h Γ ′ , Γ − Γ ′ Γ i = e (577) h Γ , Γ i = h Γ ′ , Γ i = h Γ − Γ ′ Γ , Γ i = e (578)[Γ ′ Γ Γ ′ Γ − Γ ′ − , Γ ] = e (579)[Γ ′ Γ Γ ′ Γ − Γ ′ − , Γ − Γ ′ − Γ − Γ ′ Γ Γ ′ Γ ] = e (580) h Γ , Γ i = h Γ , Γ ′ i = h Γ , Γ − Γ ′ Γ i = e (581) h Γ , Γ − Γ ′ Γ i = h Γ ′ , Γ − Γ ′ Γ i = h Γ − Γ ′ Γ , Γ − Γ ′ Γ i = e (582)[Γ ′ Γ Γ Γ − Γ ′ − , Γ − Γ ′ Γ ] = e (583)[Γ ′ Γ Γ Γ − Γ ′ − , Γ − Γ ′ − Γ − Γ ′ − Γ Γ ′ Γ Γ ′ Γ ] = e (584)Γ ′ Γ Γ ′ Γ Γ ′− Γ − Γ ′ − = Γ Γ ′ Γ − (585)Γ ′ Γ Γ ′ Γ ′ Γ ′− Γ − Γ ′ − = Γ Γ ′ Γ Γ ′ − Γ − (586)Γ ′ Γ Γ Γ Γ − Γ − Γ ′ − = Γ − Γ ′ Γ Γ ′ Γ − Γ ′ − Γ (587)Γ ′ Γ Γ Γ ′ Γ − Γ − Γ ′ − = Γ − Γ ′ Γ Γ ′ Γ Γ ′− Γ − Γ ′ − Γ . (588)We also have the following parasitic and projective relations:[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (589)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (590)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (591)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (592)Γ ′ Γ Γ ′ Γ Γ ′ Γ Γ ′ Γ Γ ′ Γ Γ ′ Γ = e. (593)By substituting (567) and equating (587) and (588), we get Γ = Γ ′ . By Lemma 3.5, we haveΓ = Γ ′ and Γ = Γ ′ . Then by Lemma 3.3, we get Γ = Γ ′ and Γ = Γ ′ . G is thus generated by { Γ i | i = 1 , . . . , } with the following relations:Γ = Γ = Γ = Γ = Γ = Γ = e (594) h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = e (595)[Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = e (596)Γ Γ Γ = Γ Γ Γ . (597)By Remark 2.8 in [24], we get also the relations[Γ , Γ Γ Γ ] = [Γ , Γ Γ Γ ] = e. (598)By Theorem 2.3 in [24] we have e G ∼ = S , therefore π ( X Gal ) is trivial.In this case, the number of branch points is 9, the number of cusps is 24, the number of nodes is24 and deg S = 12. So the first Chern number is c = 9 · c = · χ = − · B.22 X degenerates to U , Figure 42:
The arrangement of planes U , . Consider X that degenerates to U , . The branch curve S in CP is an arrangement of six lines. Weregenerate each vertex in turn and compute the group G .Vertex 2 is a 2-point. The braid monodromy corresponding to this point is: f ∆ = ( Z ′ ) Z
21 1 ′ , · Z
31 1 ′ , . f ∆ gives rise to the following relations: h Γ , Γ i = h Γ ′ , Γ i = h Γ − Γ ′ Γ , Γ i = e (599) ′ = Γ Γ ′ Γ Γ Γ − Γ ′− Γ − . (600)Vertex 3 is a 2-point. The braid monodromy corresponding to this point is: f ∆ = ( Z ′ ) Z
21 1 ′ , · Z
31 1 ′ , . f ∆ gives rise to the following relations: h Γ , Γ i = h Γ ′ , Γ i = h Γ − Γ ′ Γ , Γ i = e (601)Γ ′ = Γ Γ ′ Γ Γ Γ − Γ ′− Γ − . (602)Vertex 5 is a 2-point. The braid monodromy corresponding to this point is: f ∆ = ( Z ′ ) Z
24 4 ′ , · Z
34 4 ′ , . f ∆ gives rise to the following relations: h Γ , Γ i = h Γ ′ , Γ i = h Γ − Γ ′ Γ , Γ i = e (603)Γ ′ = Γ Γ ′ Γ Γ Γ − Γ ′− Γ − . (604)Vertex 6 is a 2-point. The braid monodromy corresponding to this point is: f ∆ = ( Z ′ ) Z
25 5 ′ , · Z
35 5 ′ , . f ∆ gives rise to the following relations: h Γ , Γ i = h Γ ′ , Γ i = h Γ − Γ ′ Γ , Γ i = e (605)Γ ′ = Γ Γ ′ Γ Γ Γ − Γ ′− Γ − . (606)Vertex 4 is an inner 4-point. Its braid monodromy appears in [8]. The corresponding relations in G are: h Γ ′ , Γ i = h Γ ′ , Γ ′ i = h Γ ′ , Γ − Γ ′ Γ i = e (607) h Γ , Γ i = h Γ ′ , Γ i = h Γ − Γ ′ Γ , Γ i = e (608)[Γ ′ Γ Γ ′ Γ − Γ ′ − , Γ ] = e (609)[Γ ′ Γ Γ ′ Γ − Γ ′ − , Γ − Γ ′ − Γ − Γ ′ Γ Γ ′ Γ ] = e (610) h Γ , Γ i = h Γ , Γ ′ i = h Γ , Γ − Γ ′ Γ i = e (611) h Γ , Γ − Γ ′ Γ i = h Γ ′ , Γ − Γ ′ Γ i = h Γ − Γ ′ Γ , Γ − Γ ′ Γ i = e (612)[Γ ′ Γ Γ Γ − Γ ′ − , Γ − Γ ′ Γ ] = e (613)[Γ ′ Γ Γ Γ − Γ ′ − , Γ − Γ ′ − Γ − Γ ′ − Γ Γ ′ Γ Γ ′ Γ ] = e (614)Γ ′ Γ Γ ′ Γ Γ ′− Γ − Γ ′ − = Γ Γ ′ Γ − (615) ′ Γ Γ ′ Γ ′ Γ ′− Γ − Γ ′ − = Γ Γ ′ Γ Γ ′ − Γ − (616)Γ ′ Γ Γ Γ Γ − Γ − Γ ′ − = Γ − Γ ′ Γ Γ ′ Γ − Γ ′ − Γ (617)Γ ′ Γ Γ Γ ′ Γ − Γ − Γ ′ − = Γ − Γ ′ Γ Γ ′ Γ Γ ′− Γ − Γ ′ − Γ . (618)We also have the following parasitic and projective relations:[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (619)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (620)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (621)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (622)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (623)Γ ′ Γ Γ ′ Γ Γ ′ Γ Γ ′ Γ Γ ′ Γ Γ ′ Γ = e. (624)As mentioned in Section 4, it is a challenging problem to determine if the kernel of the naturalsurjection from e G to S is trivial or not. This question should be addressed once the relevant algebraictools are developed. We note that the use of the computer aided algebra system MAGMA, have notyielded satisfactory results.In this case, the number of branch points is 8, the number of cusps is 24, the number of nodes is24 and deg S = 12. So the first Chern number is c = 9 · c = 5 · χ = − · B.23 X degenerates to U ,
12 3 4 56 7 612 34 5
Figure 43:
The arrangement of planes U , . Theorem B.22. If X degenerates to U , , then π ( X Gal ) is trivial. roof. The branch curve S in CP is an arrangement of six lines. We regenerate each vertex in turnand compute the group G .Vertex 1 is a 1-point that gives rise to braid Z ′ and to the following relation in G :Γ = Γ ′ . (625)Vertex 2 is a 1-point that gives rise to braid Z ′ and to the following relation in G :Γ = Γ ′ . (626)Vertex 7 is a 1-point that gives rise to braid Z ′ and to the following relation in G :Γ = Γ ′ . (627)Vertex 6 is a 2-point. The braid monodromy corresponding to this point is: f ∆ = ( Z ′ ) Z
24 4 ′ , · Z
34 4 ′ , . f ∆ gives rise to the following relations: h Γ , Γ i = h Γ ′ , Γ i = h Γ − Γ ′ Γ , Γ i = e (628)Γ ′ = Γ Γ ′ Γ Γ Γ − Γ ′− Γ − . (629)Vertex 4 is an outer 3-point. The braid monodromy corresponding to this 3-point is: f ∆ = Z
23 3 ′ , ′ · Z ′ , ′ · ( Z ′ ) Z ′ , ′ · ( Z
35 5 ′ , ) Z ′ , ′ · ( Z ′ ) Z
25 5 ′ , Z ′ , ′ . f ∆ thus gives rise to the following relations: h Γ ′ , Γ i = h Γ ′ , Γ ′ i = h Γ ′ , Γ − Γ ′ Γ i = e (630)Γ = Γ ′ Γ Γ ′ Γ − Γ ′ − (631) h Γ , Γ ′ Γ Γ ′ Γ Γ ′ − Γ − Γ ′ − i = h Γ , Γ ′ Γ Γ ′ Γ ′ Γ ′ − Γ − Γ ′ − i == h Γ , Γ ′ Γ Γ ′ Γ − Γ ′ Γ Γ ′ − Γ − Γ ′ − i = e (632)Γ ′ = Γ Γ ′ Γ Γ ′ Γ ′ Γ Γ ′ − Γ − Γ ′ − Γ Γ ′ Γ Γ ′ Γ − Γ ′ − Γ ′ − Γ − Γ ′ − Γ − (633)[Γ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ] = [Γ ′ , Γ ′ ] = e. (634)Vertex 3 is an inner 4-point. Its braid monodromy appears in [8]. The corresponding relations in G are: h Γ ′ , Γ i = h Γ ′ , Γ ′ i = h Γ ′ , Γ − Γ ′ Γ i = e (635) h Γ , Γ i = h Γ ′ , Γ i = h Γ − Γ ′ Γ , Γ i = e (636)[Γ ′ Γ Γ ′ Γ − Γ ′ − , Γ ] = e (637) Γ ′ Γ Γ ′ Γ − Γ ′ − , Γ − Γ ′ − Γ − Γ ′ Γ Γ ′ Γ ] = e (638) h Γ , Γ i = h Γ , Γ ′ i = h Γ , Γ − Γ ′ Γ i = e (639) h Γ , Γ − Γ ′ Γ i = h Γ ′ , Γ − Γ ′ Γ i = h Γ − Γ ′ Γ , Γ − Γ ′ Γ i = e (640)[Γ ′ Γ Γ Γ − Γ ′ − , Γ − Γ ′ Γ ] = e (641)[Γ ′ Γ Γ Γ − Γ ′ − , Γ − Γ ′ − Γ − Γ ′ − Γ Γ ′ Γ Γ ′ Γ ] = e (642)Γ ′ Γ Γ ′ Γ Γ ′− Γ − Γ ′ − = Γ Γ ′ Γ − (643)Γ ′ Γ Γ ′ Γ ′ Γ ′− Γ − Γ ′ − = Γ Γ ′ Γ Γ ′ − Γ − (644)Γ ′ Γ Γ Γ Γ − Γ − Γ ′ − = Γ − Γ ′ Γ Γ ′ Γ − Γ ′ − Γ (645)Γ ′ Γ Γ Γ ′ Γ − Γ − Γ ′ − = Γ − Γ ′ Γ Γ ′ Γ Γ ′− Γ − Γ ′ − Γ . (646)We also have the following parasitic and projective relations:[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (647)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (648)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (649)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (650)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (651)Γ ′ Γ Γ ′ Γ Γ ′ Γ Γ ′ Γ Γ ′ Γ Γ ′ Γ = e. (652)Substituting (625) and (626) and equating (643) and (644), we get Γ = Γ ′ . By Lemma 3.5, wehave Γ = Γ ′ . Then by Lemma 3.3, we get Γ = Γ ′ . e G is thus generated by { Γ i | i = 1 , . . . , } with the following relations:Γ = Γ = Γ = Γ = Γ = Γ = e (653) h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = e (654)[Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = e (655)Γ Γ Γ = Γ Γ Γ . (656)By Remark 2.8 in [24], we get also the relation[Γ , Γ Γ Γ ] = e. (657)By Theorem 2.3 in [24] we have e G ∼ = S , therefore π ( X Gal ) is trivial.In this case, the number of branch points is 10, the number of cusps is 21, the number of nodes is28 and deg S = 12. So the first Chern number is c = 9 · c = · χ = − · .24 X degenerates to U ∪ , Figure 44:
The arrangement of planes U ∪ , . Theorem B.23. If X degenerates to U ∪ , , then π ( X Gal ) is trivial.Proof. See Theorem 3.8.In this case, the number of branch points is 13, the number of cusps is 30, the number of nodes is36 and deg S = 14. So the first Chern number is c = 16 · c = · χ = − B.25 X degenerates to U ∪ , Figure 45:
The arrangement of planes U ∪ , . Theorem B.24. If X degenerates to U ∪ , , then π ( X Gal ) is trivial. roof. The branch curve S in CP is an arrangement of seven lines. We regenerate each vertex inturn and compute the group G .Vertex 4 is a 2-point. The braid monodromy corresponding to this point is: f ∆ = ( Z ′ ) Z
21 1 ′ , · Z
31 1 ′ , . f ∆ gives rise to the following relations: h Γ , Γ i = h Γ ′ , Γ i = h Γ − Γ ′ Γ , Γ i = e (658)Γ ′ = Γ Γ ′ Γ Γ Γ − Γ ′− Γ − . (659)Vertex 6 is a 2-point. The braid monodromy corresponding to this point is: f ∆ = ( Z ′ ) Z
22 2 ′ , · Z
32 2 ′ , . f ∆ gives rise to the following relations: h Γ , Γ i = h Γ ′ , Γ i = h Γ − Γ ′ Γ , Γ i = e (660)Γ ′ = Γ Γ ′ Γ Γ Γ − Γ ′− Γ − . (661)Vertex 2 is an inner 3-point. Its braid monodromy is f ∆ = Z ′ , ′ · Z , ′ · ( Z ′ ) Z
26 6 ′ Z ′ , ′ · ( Z ′ ) Z
26 6 ′ Z
27 7 ′ Z ′ , ′ · ( Z ′ ) Z
26 6 ′ Z , ′ · ( Z ′ ) Z
26 6 ′ Z
27 7 ′ Z , ′ . These braids give rise to the following relations in G : h Γ ′ , Γ i = h Γ ′ , Γ ′ i = h Γ ′ , Γ − Γ ′ Γ i = e (662) h Γ , Γ i = h Γ , Γ ′ i = h Γ , Γ − Γ ′ Γ i = e (663)Γ ′ Γ Γ ′ Γ Γ ′− Γ − Γ ′ − = Γ ′ (664)Γ ′ Γ Γ ′ Γ ′ Γ ′− Γ − Γ ′ − = Γ ′ Γ Γ ′ − (665)Γ ′ Γ Γ Γ Γ − Γ − Γ ′ − = Γ ′ (666)Γ ′ Γ Γ Γ ′ Γ − Γ − Γ ′ − = Γ ′ Γ Γ ′− . (667)Vertex 3 is an outer 3-point. The braid monodromy corresponding to this 3-point is: f ∆ = Z
24 4 ′ , ′ · Z ′ , ′ · ( Z ′ ) Z ′ , ′ · ( Z
35 5 ′ , ) Z ′ , ′ · ( Z ′ ) Z
25 5 ′ , Z ′ , ′ . f ∆ thus gives rise to the following relations: h Γ ′ , Γ i = h Γ ′ , Γ ′ i = h Γ ′ , Γ − Γ ′ Γ i = e (668)Γ = Γ ′ Γ Γ ′ Γ − Γ ′ − (669) Γ , Γ ′ Γ Γ ′ Γ Γ ′ − Γ − Γ ′ − i = h Γ , Γ ′ Γ Γ ′ Γ ′ Γ ′ − Γ − Γ ′ − i == h Γ , Γ ′ Γ Γ ′ Γ − Γ ′ Γ Γ ′ − Γ − Γ ′ − i = e (670)Γ ′ = Γ Γ ′ Γ Γ ′ Γ ′ Γ Γ ′ − Γ − Γ ′ − Γ Γ ′ Γ Γ ′ Γ − Γ ′ − Γ ′ − Γ − Γ ′ − Γ − (671)[Γ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ] = [Γ ′ , Γ ′ ] = e. (672)Vertex 1 is an inner 4-point. Its braid monodromy appears in [8]. The corresponding relations in G are: h Γ ′ , Γ i = h Γ ′ , Γ ′ i = h Γ ′ , Γ − Γ ′ Γ i = e (673) h Γ , Γ i = h Γ ′ , Γ i = h Γ − Γ ′ Γ , Γ i = e (674)[Γ ′ Γ Γ ′ Γ − Γ ′ − , Γ ] = e (675)[Γ ′ Γ Γ ′ Γ − Γ ′ − , Γ − Γ ′ − Γ − Γ ′ Γ Γ ′ Γ ] = e (676) h Γ , Γ i = h Γ , Γ ′ i = h Γ , Γ − Γ ′ Γ i = e (677) h Γ , Γ − Γ ′ Γ i = h Γ ′ , Γ − Γ ′ Γ i = h Γ − Γ ′ Γ , Γ − Γ ′ Γ i = e (678)[Γ ′ Γ Γ Γ − Γ ′ − , Γ − Γ ′ Γ ] = e (679)[Γ ′ Γ Γ Γ − Γ ′ − , Γ − Γ ′ − Γ − Γ ′ − Γ Γ ′ Γ Γ ′ Γ ] = e (680)Γ ′ Γ Γ ′ Γ Γ ′− Γ − Γ ′ − = Γ Γ ′ Γ − (681)Γ ′ Γ Γ ′ Γ ′ Γ ′− Γ − Γ ′ − = Γ Γ ′ Γ Γ ′ − Γ − (682)Γ ′ Γ Γ Γ Γ − Γ − Γ ′ − = Γ − Γ ′ Γ Γ ′ Γ − Γ ′ − Γ (683)Γ ′ Γ Γ Γ ′ Γ − Γ − Γ ′ − = Γ − Γ ′ Γ Γ ′ Γ Γ ′− Γ − Γ ′ − Γ . (684)We also have the following parasitic and projective relations:[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (685)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (686)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (687)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (688)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (689)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (690)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (691)Γ ′ Γ Γ ′ Γ Γ ′ Γ Γ ′ Γ Γ ′ Γ Γ ′ Γ Γ ′ Γ = e. (692) y Lemma 3.4, we have Γ = Γ ′ . Using this result and equating (681) and (682), we get Γ = Γ ′ .By Lemma 3.3, we have Γ = Γ ′ . Then by Lemma 3.5, we get Γ = Γ ′ and Γ = Γ ′ . Using Lemma3.4, we get Γ = Γ ′ . And again by Lemma 3.3 we have Γ = Γ ′ . e G is thus generated by { Γ i | i = 1 , . . . , } with the following relations:Γ = Γ = Γ = Γ = Γ = Γ = Γ = e (693) h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = (694)= h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = e [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = (695)= [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = e Γ = Γ Γ Γ (696)Γ Γ Γ = Γ Γ Γ . (697)By Remark 2.8 in [24], we get also the relations[Γ , Γ Γ Γ ] = [Γ , Γ Γ Γ ] = [Γ , Γ Γ Γ ] = e. (698)By Theorem 2.3 in [24] we have e G ∼ = S , therefore π ( X Gal ) is trivial.In this case, the number of branch points is 12, the number of cusps is 30, the number of nodes is36 and deg S = 14. So the first Chern number is c = 16 · c = 9 · χ = − · B.26 X degenerates to U ∪ Figure 46:
The arrangement of planes U ∪ . Theorem B.25. If X degenerates to U ∪ , then π ( X Gal ) is trivial. roof. The branch curve S in CP is an arrangement of seven lines. We regenerate each vertex inturn and compute the group G .Vertex 1 is a 1-point that gives rise to braid Z ′ and to the following relation in G :Γ = Γ ′ . (699)Vertex 6 is a 1-point that gives rise to braid Z ′ and to the following relation in G :Γ = Γ ′ . (700)Vertex 3 is a 2-point. The braid monodromy corresponding to this point is: f ∆ = ( Z ′ ) Z
25 5 ′ , · Z
35 5 ′ , . f ∆ gives rise to the following relations: h Γ , Γ i = h Γ ′ , Γ i = h Γ − Γ ′ Γ , Γ i = e (701)Γ ′ = Γ Γ ′ Γ Γ Γ − Γ ′− Γ − . (702)Vertex 4 is a 2-point. The braid monodromy corresponding to this point is: f ∆ = ( Z ′ ) Z
21 1 ′ , · Z
31 1 ′ , . f ∆ gives rise to the following relations: h Γ , Γ i = h Γ ′ , Γ i = h Γ − Γ ′ Γ , Γ i = e (703)Γ ′ = Γ Γ ′ Γ Γ Γ − Γ ′− Γ − . (704)Vertex 2 is an inner 4-point. Its braid monodromy appears in [8]. The corresponding relations in G are: h Γ ′ , Γ i = h Γ ′ , Γ ′ i = h Γ ′ , Γ − Γ ′ Γ i = e (705) h Γ , Γ i = h Γ ′ , Γ i = h Γ − Γ ′ Γ , Γ i = e (706)[Γ ′ Γ Γ ′ Γ − Γ ′ − , Γ ] = e (707)[Γ ′ Γ Γ ′ Γ − Γ ′ − , Γ − Γ ′ − Γ − Γ ′ Γ Γ ′ Γ ] = e (708) h Γ , Γ i = h Γ , Γ ′ i = h Γ , Γ − Γ ′ Γ i = e (709) h Γ , Γ − Γ ′ Γ i = h Γ ′ , Γ − Γ ′ Γ i = h Γ − Γ ′ Γ , Γ − Γ ′ Γ i = e (710)[Γ ′ Γ Γ Γ − Γ ′ − , Γ − Γ ′ Γ ] = e (711)[Γ ′ Γ Γ Γ − Γ ′ − , Γ − Γ ′ − Γ − Γ ′ − Γ Γ ′ Γ Γ ′ Γ ] = e (712)Γ ′ Γ Γ ′ Γ Γ ′− Γ − Γ ′ − = Γ Γ ′ Γ − (713) ′ Γ Γ ′ Γ ′ Γ ′− Γ − Γ ′ − = Γ Γ ′ Γ Γ ′ − Γ − (714)Γ ′ Γ Γ Γ Γ − Γ − Γ ′ − = Γ − Γ ′ Γ Γ ′ Γ − Γ ′ − Γ (715)Γ ′ Γ Γ Γ ′ Γ − Γ − Γ ′ − = Γ − Γ ′ Γ Γ ′ Γ Γ ′− Γ − Γ ′ − Γ . (716)Vertex 5 is an inner 4-point. Its braid monodromy appears in [8]. The corresponding relations in G are: h Γ ′ , Γ i = h Γ ′ , Γ ′ i = h Γ ′ , Γ − Γ ′ Γ i = e (717) h Γ , Γ i = h Γ ′ , Γ i = h Γ − Γ ′ Γ , Γ i = e (718)[Γ ′ Γ Γ ′ Γ − Γ ′ − , Γ ] = e (719)[Γ ′ Γ Γ ′ Γ − Γ ′ − , Γ − Γ ′ − Γ − Γ ′ Γ Γ ′ Γ ] = e (720) h Γ , Γ i = h Γ , Γ ′ i = h Γ , Γ − Γ ′ Γ i = e (721) h Γ , Γ − Γ ′ Γ i = h Γ ′ , Γ − Γ ′ Γ i = h Γ − Γ ′ Γ , Γ − Γ ′ Γ i = e (722)[Γ ′ Γ Γ Γ − Γ ′ − , Γ − Γ ′ Γ ] = e (723)[Γ ′ Γ Γ Γ − Γ ′ − , Γ − Γ ′ − Γ − Γ ′ − Γ Γ ′ Γ Γ ′ Γ ] = e (724)Γ ′ Γ Γ ′ Γ Γ ′− Γ − Γ ′ − = Γ Γ ′ Γ − (725)Γ ′ Γ Γ ′ Γ ′ Γ ′− Γ − Γ ′ − = Γ Γ ′ Γ Γ ′ − Γ − (726)Γ ′ Γ Γ Γ Γ − Γ − Γ ′ − = Γ − Γ ′ Γ Γ ′ Γ − Γ ′ − Γ (727)Γ ′ Γ Γ Γ ′ Γ − Γ − Γ ′ − = Γ − Γ ′ Γ Γ ′ Γ Γ ′− Γ − Γ ′ − Γ . (728)We also have the following parasitic and projective relations:[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (729)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (730)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (731)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (732)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (733)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (734)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (735)Γ ′ Γ Γ ′ Γ Γ ′ Γ Γ ′ Γ Γ ′ Γ Γ ′ Γ Γ ′ Γ = e. (736)Substituting (699) and equating (713) and (714), we get Γ = Γ ′ .Using computer algebra system (we used MAGMA), we can express all the Γ ′ i ’s by { Γ j | j =1 , . . . , } and get all the defining relations of S . Then it is a straight forward verification that allthe relations in G hold in S so G/ h Γ i i ∼ = S , and π ( X Gal ) is trivial. n this case, the number of branch points is 12, the number of cusps is 30, the number of nodes is36 and deg S = 14. So the first Chern number is c = 16 · c = 9 · χ = − · B.27 X degenerates to U
12 3 45 67 1 52 346
Figure 47:
The arrangement of planes U . Theorem B.26. If X degenerates to U , then π ( X Gal ) is trivial.Proof. The branch curve S in CP is an arrangement of six lines. We regenerate each vertex in turnand compute the group G .Vertex 1 is a 1-point that gives rise to braid Z ′ and to the following relation in G :Γ = Γ ′ . (737)Vertex 2 is a 1-point that gives rise to braid Z ′ and to the following relation in G :Γ = Γ ′ . (738)Vertex 4 is a 1-point that gives rise to braid Z ′ and to the following relation in G :Γ = Γ ′ . (739)Vertex 5 is a 2-point. The braid monodromy corresponding to this point is: f ∆ = ( Z ′ ) Z
24 4 ′ , · Z
34 4 ′ , . f ∆ gives rise to the following relations: h Γ , Γ i = h Γ ′ , Γ i = h Γ − Γ ′ Γ , Γ i = e (740) ′ = Γ Γ ′ Γ Γ Γ − Γ ′− Γ − . (741)Vertex 6 is a 2-point. The braid monodromy corresponding to this point is: f ∆ = ( Z ′ ) Z
25 5 ′ , · Z
35 5 ′ , . f ∆ gives rise to the following relations: h Γ , Γ i = h Γ ′ , Γ i = h Γ − Γ ′ Γ , Γ i = e (742)Γ ′ = Γ Γ ′ Γ Γ Γ − Γ ′− Γ − . (743)Vertex 3 is a 5-point. According to [16, Corollary 2.5], the braid monodromy corresponding to ityields the following relations in G : [Γ , Γ ] = [Γ ′ , Γ ] = e (744) h Γ ′ , Γ i = h Γ ′ , Γ ′ i = h Γ ′ , Γ − Γ ′ Γ i = e (745) h Γ , Γ i = h Γ ′ , Γ i = h Γ − Γ ′ Γ , Γ i = e (746)[Γ Γ Γ − , Γ ′ Γ Γ ′ Γ − Γ ′ − ] = [Γ Γ ′ Γ − , Γ ′ Γ Γ ′ Γ − Γ ′ − ] = e (747)Γ Γ ′ Γ Γ Γ − Γ ′ − Γ − = Γ ′ Γ Γ ′ Γ − Γ ′ − (748)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ Γ Γ ′ Γ − Γ ′ − ] = [Γ ′ , Γ ′ Γ Γ ′ Γ − Γ ′ − ] = e (749) h Γ ′ , Γ i = h Γ ′ , Γ ′ i = h Γ ′ , Γ − Γ ′ Γ i = e (750) h Γ Γ Γ − , Γ i = h Γ Γ ′ Γ − , Γ i = h Γ Γ − Γ ′ Γ Γ − , Γ i = e (751)Γ ′ Γ Γ ′ Γ Γ ′− Γ − Γ ′ − = Γ − Γ Γ Γ ′ Γ − Γ − Γ (752)Γ ′ Γ Γ ′ Γ ′ Γ ′− Γ − Γ ′ − = Γ − Γ Γ Γ ′ Γ Γ ′− Γ − Γ − Γ (753)[Γ ′ Γ Γ ′ Γ − Γ ′ − , Γ − Γ Γ ] = e (754)[Γ ′ Γ Γ ′ Γ Γ ′ Γ − Γ ′ − Γ − Γ ′− , Γ − Γ − Γ ′ Γ Γ ] = e (755) h Γ , Γ i = h Γ , Γ ′ i = h Γ , Γ − Γ ′ Γ i = e (756) h Γ Γ Γ − , Γ − Γ ′ Γ i = h Γ Γ ′ Γ − , Γ − Γ ′ Γ i = h Γ Γ − Γ ′ Γ Γ − , Γ − Γ ′ Γ i = e (757)Γ ′ Γ Γ Γ Γ − Γ − Γ ′ − = Γ − Γ − Γ ′ Γ Γ Γ ′ Γ − Γ − Γ ′ − Γ Γ (758)Γ ′ Γ Γ Γ ′ Γ − Γ − Γ ′ − = Γ − Γ − Γ ′ Γ Γ Γ ′ Γ Γ ′− Γ − Γ − Γ ′ − Γ Γ (759)[Γ ′ Γ Γ Γ − Γ ′ − , Γ − Γ − Γ ′ Γ Γ ] = e, (760)[Γ ′ Γ Γ ′ Γ Γ Γ − Γ ′ − Γ − Γ ′− , Γ − Γ − Γ ′ − Γ Γ ′ Γ Γ ] = e. (761) e also have the following parasitic and projective relations:[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (762)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (763)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (764)Γ ′ Γ Γ ′ Γ Γ ′ Γ Γ ′ Γ Γ ′ Γ Γ ′ Γ = e. (765)Using (737), (738), (739), (751), (757) and equating (752) and (758), we get Γ = Γ ′ . By Lemma3.3, Γ = Γ ′ and Γ = Γ ′ . e G is thus generated by { Γ i | i = 1 , . . . , } with the following relations:Γ = Γ = Γ = Γ = Γ = Γ = e (766) h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = e (767)[Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = e (768)Γ Γ Γ = Γ Γ Γ Γ Γ . (769)By Remark 2.8 in [24], we get also the relation[Γ , Γ Γ Γ ] = e. (770)By Theorem 2.3 in [24] we have e G ∼ = S , therefore π ( X Gal ) is trivial.In this case, the number of branch points is 10, the number of cusps is 24, the number of nodes is24 and deg S = 12. So the first Chern number is c = 9 · c = 6 · χ = − B.28 X degenerates to U ∪ Figure 48:
The arrangement of planes U ∪ . Theorem B.27. If X degenerates to U ∪ , then π ( X Gal ) is trivial. roof. The branch curve S in CP is an arrangement of seven lines. We regenerate each vertex inturn and compute the group G .Vertex 1 is a 1-point that gives rise to braid Z ′ and to the following relation in G :Γ = Γ ′ . (771)Vertex 2 is a 1-point that gives rise to braid Z ′ and to the following relation in G :Γ = Γ ′ . (772)Vertex 5 is a 2-point. The braid monodromy corresponding to this point is: f ∆ = ( Z ′ ) Z
25 5 ′ , · Z
35 5 ′ , . f ∆ gives rise to the following relations: h Γ , Γ i = h Γ ′ , Γ i = h Γ − Γ ′ Γ , Γ i = e (773)Γ ′ = Γ Γ ′ Γ Γ Γ − Γ ′− Γ − . (774)Vertex 6 is a 2-point. The braid monodromy corresponding to this point is: f ∆ = ( Z ′ ) Z
22 2 ′ , · Z
32 2 ′ , . f ∆ gives rise to the following relations: h Γ , Γ i = h Γ ′ , Γ i = h Γ − Γ ′ Γ , Γ i = e (775)Γ ′ = Γ Γ ′ Γ Γ Γ − Γ ′− Γ − . (776)Vertex 4 is an inner 3-point. Its braid monodromy is f ∆ = Z ′ , ′ · Z , ′ · ( Z ′ ) Z
26 6 ′ Z ′ , ′ · ( Z ′ ) Z
26 6 ′ Z
27 7 ′ Z ′ , ′ · ( Z ′ ) Z
26 6 ′ Z , ′ · ( Z ′ ) Z
26 6 ′ Z
27 7 ′ Z , ′ . These braids give rise to the following relations in G : h Γ ′ , Γ i = h Γ ′ , Γ ′ i = h Γ ′ , Γ − Γ ′ Γ i = e (777) h Γ , Γ i = h Γ , Γ ′ i = h Γ , Γ − Γ ′ Γ i = e (778)Γ ′ Γ Γ ′ Γ Γ ′− Γ − Γ ′ − = Γ ′ (779)Γ ′ Γ Γ ′ Γ ′ Γ ′− Γ − Γ ′ − = Γ ′ Γ Γ ′ − (780)Γ ′ Γ Γ Γ Γ − Γ − Γ ′ − = Γ ′ (781)Γ ′ Γ Γ Γ ′ Γ − Γ − Γ ′ − = Γ ′ Γ Γ ′− . (782) ertex 3 is a 5-point. According to [16, Corollary 2.5], the braid monodromy corresponding to ityields the following relations in G : [Γ , Γ ] = [Γ ′ , Γ ] = e (783) h Γ ′ , Γ i = h Γ ′ , Γ ′ i = h Γ ′ , Γ − Γ ′ Γ i = e (784) h Γ , Γ i = h Γ ′ , Γ i = h Γ − Γ ′ Γ , Γ i = e (785)[Γ Γ Γ − , Γ ′ Γ Γ ′ Γ − Γ ′ − ] = [Γ Γ ′ Γ − , Γ ′ Γ Γ ′ Γ − Γ ′ − ] = e (786)Γ Γ ′ Γ Γ Γ − Γ ′ − Γ − = Γ ′ Γ Γ ′ Γ − Γ ′ − (787)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ Γ Γ ′ Γ − Γ ′ − ] = [Γ ′ , Γ ′ Γ Γ ′ Γ − Γ ′ − ] = e (788) h Γ ′ , Γ i = h Γ ′ , Γ ′ i = h Γ ′ , Γ − Γ ′ Γ i = e (789) h Γ Γ Γ − , Γ i = h Γ Γ ′ Γ − , Γ i = h Γ Γ − Γ ′ Γ Γ − , Γ i = e (790)Γ ′ Γ Γ ′ Γ Γ ′− Γ − Γ ′ − = Γ − Γ Γ Γ ′ Γ − Γ − Γ (791)Γ ′ Γ Γ ′ Γ ′ Γ ′− Γ − Γ ′ − = Γ − Γ Γ Γ ′ Γ Γ ′− Γ − Γ − Γ (792)[Γ ′ Γ Γ ′ Γ − Γ ′ − , Γ − Γ Γ ] = e (793)[Γ ′ Γ Γ ′ Γ Γ ′ Γ − Γ ′ − Γ − Γ ′− , Γ − Γ − Γ ′ Γ Γ ] = e (794) h Γ , Γ i = h Γ , Γ ′ i = h Γ , Γ − Γ ′ Γ i = e (795) h Γ Γ Γ − , Γ − Γ ′ Γ i = h Γ Γ ′ Γ − , Γ − Γ ′ Γ i = h Γ Γ − Γ ′ Γ Γ − , Γ − Γ ′ Γ i = e (796)Γ ′ Γ Γ Γ Γ − Γ − Γ ′ − = Γ − Γ − Γ ′ Γ Γ Γ ′ Γ − Γ − Γ ′ − Γ Γ (797)Γ ′ Γ Γ Γ ′ Γ − Γ − Γ ′ − = Γ − Γ − Γ ′ Γ Γ Γ ′ Γ Γ ′− Γ − Γ − Γ ′ − Γ Γ (798)[Γ ′ Γ Γ Γ − Γ ′ − , Γ − Γ − Γ ′ Γ Γ ] = e, (799)[Γ ′ Γ Γ ′ Γ Γ Γ − Γ ′ − Γ − Γ ′− , Γ − Γ − Γ ′ − Γ Γ ′ Γ Γ ] = e. (800)We also have the following parasitic and projective relations:[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (801)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (802)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (803)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (804)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (805)[Γ , Γ ] = [Γ ′ , Γ ] = [Γ , Γ ′ ] = [Γ ′ , Γ ′ ] = e (806)Γ ′ Γ Γ ′ Γ Γ ′ Γ Γ ′ Γ Γ ′ Γ Γ ′ Γ Γ ′ Γ = e. (807) ubstituting (771), (772) and Equating (791) and (792), we get Γ = Γ ′ . By Lemma 3.3, Γ = Γ ′ .Then by Lemma (3.4) we get Γ = Γ ′ and Γ = Γ ′ . Then again by Lemma (3.3) we have Γ = Γ ′ . e G is thus generated by { Γ i | i = 1 , . . . , } with the following relations:Γ = Γ = Γ = Γ = Γ = Γ = Γ = e (808) h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = h Γ , Γ i == h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = h Γ , Γ i = e (809)[Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] == [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = [Γ , Γ ] = e (810)Γ = Γ Γ Γ (811)Γ Γ Γ = Γ Γ Γ Γ Γ . (812)By Remark 2.8 in [24], we get also the relations[Γ , Γ Γ Γ ] = [Γ , Γ Γ Γ ] = e. (813)By Theorem 2.3 in [24] we have e G ∼ = S , therefore π ( X Gal ) is trivial.In this case, the number of branch points is 13, the number of cusps is 30, the number of nodes is36 and deg S = 14. So the first Chern number is c = 16 · c = · χ = − B.29 X degenerates to U Figure 49:
The arrangement of planes U . Theorem B.28. If X degenerates to U , then π ( X Gal ) is trivial.Proof. See [5, Theorem 6].In this case, the number of branch points is 12, the number of cusps is 24, the number of nodes is24 and deg S = 12. So the first Chern number is c = 9 · c = 7 · χ = −6!.