BRAID MONODROMY FACTORIZATION FOR A NON-PRIME K3 SURFACE BRANCH CURVE. 1. Overview

Transcription

1 BRAID MONODROMY FACTORIZATION FOR A NON-PRIME K SURFACE BRANCH CURVE AMRAM MEIRAV, CILIBERTO CIRO, MIRANDA RICK AND TEICHER MINA Abstract This paper is the second in a series The first [5] describes pillow degenerations of a K surface with genus g In this paper we study the (, )-pillow degeneration of a non-prime K surface and the braid monodromy of the branch curve of the surface with respect to a generic projection onto CP In [4] we study the fundamental groups of the complement of the branch curve and of the corresponding Galois cover of the surface Overview Given a projective surface and a generic projection to the plane, the fundamental group of the complement of the branch curve is one of the most important invariants, of either the branch curve or the surface itself Our goal is to compute this group and the fundamental group of the Galois cover (which is a quotient) In this paper we deal with a non-prime K surface of degree 6 which is embedded in P 9 In order to compute the above groups we degenerate the surface into a union of 6 planes, then we project it onto CP to get a degenerated branch curve S 0 from which one can compute the braid monodromy and the branch curve S of K We then regenerate the curve to obtain the branch curve of our surface With this information we will be able to apply the van Kampen Theorem (see []) and the regeneration rules (see [6]) to get presentations for the relevant fundamental groups The idea of using degenerations appears already in [], [4], [6], [] and [5] Note that in [5] similar degenerations for K surfaces were computed precisely by the last three authors Partially supported by the DAAD fellowship (Germany), the Golda Meir postdoctoral fellowship (the Einstein mathematics institute, Hebrew university, Jerusalem), the Emmy Noether Research Institute for Mathematics (center of the Minerva Foundation of Germany), the Excellency Center Group Theoretic Methods in the Study of Algebraic Varieties of the Israel Science Foundation, and EAGER (EU network, HPRN-CT ) Date: December 4, 005

2 who refer to it as a pillow degeneration; the degeneration used in this paper is an example of a pillow degeneration The study of braids had been initiated in [] and later in [] and [], but the braid monodromy technique was first presented by Moishezon-Teicher in [7], [] and [] Many examples of the braid monodromy have been already computed, for example in [], [7], [] In [] one can find a description of computations of braid monodromy and the fundamental group π (CP S) of the complement of the branch curve of a surface In this work we encounter new type of singularities (-points, see Section 4) which have not been handled before, whose analysis is necessary to give the precise computations of the braid monodromy Moreover, since the monodromies are quite hard to follow, we try to present them first in a precise way algebraically, followed then by figures In a forthcoming paper [4] we deal with the computation of the fundamental group of the Galois cover of the surface The paper is divided as follows In Section we give the definition of the degeneration and explain briefly the computations from [5] In Section we review the general setup and the notion of the braid monodromy In Section 4 we recover the relevant properties of the branch curve S of the generic projection to the plane of the K surface by using regeneration techniques Section 5 states the braid monodromy factorization of 48 of S and its related invariance properties Degeneration of a K surface Let us start this section with recalling the definition of degeneration from [5] Definition Projective degeneration Let be the unit disc, and suppose that k : Y CP n and k : X CP n are projective embeddings We say that k is a projective degeneration of k if there exist a flat family π : V, and an embedding F : V CP n, such that F composed with the first projection is π, and: (a) π (0) X; (b) there is a t 0 0 in such that π (t 0 ) Y ; (c) the family V π (0) 0 is smooth; (d) restricted to π (0), F = 0 k under the identification of π (0) with X; (e) restricted to π (t 0 ), F = t 0 k under the identification of π (t 0 ) with Y

3 The example we will be working with is a total degeneration of a K surface of genus g in P g This means that this surface degenerates to a union of g planes This total degeneration of the surface is called a pillow degeneration, see [5] for details This degeneration can be viewed as two rectangular arrays of planes, joined along their boundary Fix two integers a and b at least and set g = ab + The K surface degenerates to a union of g = 4ab planes The degenerated object contains the g+ = ab+ coordinate points of the ambient space P g Each one of the planes is obtained as the span of three of these points The boundary of the two rectangular arrays of planes contains a + b lines, where the identification is taking place Taking a non-prime K surface of degree 6 in P g and fixing a = b =, we get a degeneration (K) 0 It is a union of 6 planes, as depicted in Figure (the top and the bottom parts of the pillow identify along the boundaries of the two configurations) We quote the main result from [5] Theorem The (, )-pillow degeneration (K) 0 is a total degeneration of the smooth K surface of degree 6 in P 9 The smooth K surface is a re-embedding of a quartic surface in P via the linear system of quadrics TOP BOTTOM Figure The (, )-pillow degeneration For the regeneration process we have to fix a numeration of vertices and lines, which was chosen systematically in [5] The boundary points are labeled clockwise from to 8, while the interior point on the top is 9 and on the bottom is 0 The 6 planes meet each other along a total of 4 lines, each joining two of the 0 coordinate points The numbering of the vertices induces a lexicographic numbering of the lines as follows If L has endpoints a < b

4 and M has endpoints c < d then L < M if b < d or d = b and a < c This gives a total ordering of the lines, which we interpret as a numbering from to 4, as shown in Figure Note that the coordinate points,, 5, 7 at the corners are each contained in three distinct planes, while all other points are each contained in six planes We call these two types - points and 6-points respectively The regeneration and subsequent braid computations in a neighbourhood of the -points is not treated in previous works; hence we study them here in detail But 6-points were treated in [], [7], [] and [6] Since we have similar computations, we present just the resulting braids A general projection f 0 : (K) 0 CP will also be a degeneration of a general such projection of the smooth K surface Under f 0, each of the 6 planes is mapped isomorphically to CP The ramification locus R 0 of f 0 is the closed subset of (K) 0, where f 0 is not a local isomorphism Here R 0 is exactly the 4 lines Let S 0 = f 0 (R 0 ) be the degenerated branch curve; it is a line arrangement, composed of the images of the 4 lines (each counted twice and each of which is a line in the plane) The braid monodromy notion Consider the following setting (Figure ) S is an algebraic curve in C, with p = deg(s) Let π : C C be a generic projection on the first coordinate Define the fiber K(x) = {y (x, y) S} in S over a fixed point x, projected to the y-axis Define N = {x #K(x) < p} and M = {s S π s is not étale at s}; note that π(m ) = N Let {A j } q j= be the set of points of M and N = {x j } q j= their projection on the x-axis Recall that π is generic, so we assume that #(π (x) M ) = for every x N Let E (resp D) be a closed disk on the x-axis (resp the y-axis), such that M E D and N Int(E) We choose u E a real point far enough from the set N, x << u for every x N Define C u = π (u) and number the points of K = C u S as {,,p} We construct a g-base for the fundamental group π (E N, u) Take a set of paths {γ j } q j= which connect u with the points {x j } q j= of N Now encircle each x j with a small oriented counterclockwise circle c j Denote the path segment from u to the boundary of this circle as γ j We define an element (a loop) in the g-base as δ j = γ jc j γ j Let B p [D, K] be the braid group, and let H,, H p be its frame (for complete definitions, see [, Section

5 π S C u Cu S C N u Figure General setting III]) The braid monodromy of S [] is a map ϕ : π (E N, u) B p [D, K] defined as follows: every loop in E N starting at u has liftings to a system of p paths in (E N) D starting at each point of K =,, p Projecting them to D we get p paths in D defining a motion {(t),, p(t)} (for 0 t ) of p points in D starting and ending at K This motion defines a braid in B p [D, K] By the Artin Theorem [], for j =,,q, there exists a halftwist Z j B p [D, K] and ǫ j Z, such that ϕ(δ j ) = Z ǫ j j, where Z j is a halftwist and ǫ j =, or (for an ordinary branch point, a node, or a cusp respectively) We explain now how to get this Z j Definition A curve S is called an almost real curve if () N E R, () N E E, () x E R N, #(K R)(x) p, (4) x N, #π (x) M =, (5) The singularities can be (a) a branch point, topologically equivalent to y + x = 0, (b) a branch point, topologically equivalent to y x = 0, (c) a tangent point (a line is tangent to a conic) (d) an intersection of m smooth branches, transversal to each other We deal with curves, which have the above property We explain how to get the braid monodromy around each singularity in S Let A j be a singularity in S and its projection by π to the x-axis is x j We choose a point x j next to x j, such that π (x j ) is a typical fiber If A j is (b), (c) or (d), then x j is on the right side of x j If A j is (a), then x j is on the left side of x j (the typical fiber in case (a), which is on the left side of this singularity, intersects

6 the conic in two real points) We encircle A j with a very small circle in a way that the typical fiber π (x j ) intersects the circle in two points, say a, b We fix a skeleton ξ x which j connects a and b, and denote it as < a, b > The Lefschetz diffeomorphism Ψ ([, Subsection 95]) allows us to get a resulting skeleton (ξ x j )Ψ in the typical fiber C u This one defines a motion of its two endpoints This motion induces a halftwist Z j = < (ξ x j )Ψ > As above, ϕ(δ j ) = < (ξ x j )Ψ > ǫ j The braid monodromy factorization associated to S is q p = ϕ(δ j ) j= 4 The branch curve S 0 and its regeneration 4 The branch curve S 0 The degenerated object (K) 0 has {j} 0 j= as vertices By the projection f 0 : K 0 CP, we get a line arrangement S 0 = 4 i, and the projections f 0 (j) = j are singularities of S 0 The -points are,, 5, 7 and the 6-points are, 4, 6, 8, 9, 0 In the following subsection we regenerate in neighborhoods of these singularities The monodromies will be given in the next section Besides these singularities in S 0, there are parasitic intersections We compute them now In (K) 0 there are lines which do not intersect When projecting (K) 0 onto CP, they will intersect Denote each line in (K) 0 as a pair (by its two end vertices) Let u be a point, u / S 0, such that #C u = 4, and q i = i C u a real point Take two non-intersecting lines p = (i, k) and t = (j, l) in (K) 0 i= Notation 4 We denote by Z ij (resp Zij ) the counterclockwise halftwist of i and j along a path below (resp above) the real axis If P is a set of points between i and j, (P) Z ij denotes the path from i to j going above the points in P and below the points not in P Conjugation of braids is defined as a b = b ab

7 We compute first the products D t := p<t p t= Z pt as explained in [, Theorem IX] D =D = D = Id, D 4 = Z 4 () D 8 = D = 5 i= i=4 i 0 D 7 = D = i= i 8 Z i 8, D 9 = 8 i= Z i, D 4 = Z i 7, D 8 = () (6) 8 i= i 6,5, D 5 = Z i 9, D 0 = i= i,4 8 i= Z i, D = (9)(0) Z i 4 () i= 8 i=4, D 5 = Z i 5, D 6 = Z i 0 (9) Z i 8, D 9 = () (7) 8 i= i 5,7 Define the parasitic intersection braids as i= i 6, D = 8 i= i 9 Z i, D = (9) () () Cj = j t D t, i= 5 i= Z i 6, D 7 = Z i (9)(0) Z i 5, D 6 = ()(4) Z i 9, D 0 = 8 i= i 6 8,,6 4 i=, D = i= i 6 8, 8 i=4 i 0, Z i 7 (6) 7 i= Z i 6, () (5) Z i 0 (9) Z i, D 4 = (9) (), 7 i= i 9, Z i, (9) () Z i 4 (9) () where j is the smallest endpoint in the line t In our case, C = D D 9 D 9, C = D D D 0 D D 0, C = D 4 D 4, C4 = D 5 D 6 D 5 D, C5 = D 7 D, C6 = D 8 D D 6 D, C7 = D D 7, C8 = D 8 D 4, C9 = C 0 = Id 4 The regeneration of S 0 The degenerated branch curve S 0 of (K) 0 has degree 4 However each of the 4 lines of S 0 should be counted as a double line in the scheme-theoretic branch locus, since it arises from a line of nodes Another way to see this is to note that the regeneration of (K) 0 induces a regeneration of S 0 in such a way that each point, say c, on the typical fiber is replaced by two close points c, c The curve S 0 has -points, 6-points and parasitic intersections In the forthcoming subsections we explain how to regenerate in their neighbourhoods The resulting branch curve S will have degree 48 4 Regeneration of -points The -points in S 0 are,, 5, 7, see Figure The regeneration is divided into steps We explain each step in two levels, the one of the surface and the second one of the branch curve

8 In the surface level, each diagonal is replaced by a conic by a partial regeneration Focusing on a -point, we have a partial regeneration of two of the planes to a quadric surface We get one quadric and one plane, which is tangent to the quadric The plane and the quadric meet along two lines (one from each ruling of the quadric) For the regeneration in the branch curve level, we need the following lemma from [6] Lemma 5 Let V be a projective algebraic surface, and D be a curve in V Let f : V CP be a generic projection Let S CP, S V be the branch curve of f and the corresponding ramification curve Assume S intersects D in α Let D = f(d ) and α = f(α ) Assume that there exist neighbourhoods of α and α, such that f S and f D are isomorphisms Then D is tangent to S at α In the branch curve level, we have two double lines (coming from the intersection of the plane and the quadric) and one conic (coming from the branching of the quadric over the plane) According to the above lemma, the conic is tangent to each of the two double lines As far as the branch points go, one of the two branch points of the conic is far away from the -point, the other one is close to the -point, see for example Figure Figure Partial regeneration in a neighbourhood of the -point Now in the next step of the regeneration, in the curve level, we use regeneration Lemmas from [] The two tangent points regenerate to three cusps each (giving a total of six) and the intersection point of the two double lines gives eight more branch points One can think of this as first giving four nodes, then each node is giving two branch points (like the line-crossing regenerating to a conic) In the surface level it means that we get a smooth surface which locally looks like a cubic in CP (degenerating to a triple of planes) 4 Regeneration of 6-points The 6-points in S 0 are, 4, 6, 8, 9, 0 Each of these points is the projection to the plane of a cone over a cycle of independent lines spanning a CP 5 Hence originally the surface lives in CP 6 and consists of six planes through a point

9 We focus on a neighbourhood around in S 0 The first regeneration consists in smoothing out the lines and 0 and therefore replacing four of these pairwise adjacent but opposite planes with two quadrics, see Figure Figure 4 Partial regeneration in a neighbourhood of the 6-point Then one smooths out the lines and 0 This produces two cubic rational normal scrolls meeting along the lines and Now the union of these lines on each scroll is homologous to a non singular conic, so we can make the two scrolls meeting along a smooth conic (in this way and are replaced by a smooth double conic) We visualize a purely local figure (Figure 5), in order to understand this step Finally one smooths the conic too and arrives to a Del-Pezzo sextic in CP 6 A similar regeneration occurs in the neighbourhoods of the other 6-points 5 The braid monodromy factorization 48 In Section 4 we explained how to get the branch curve S The degree of S is 48 We want to compute the braid monodromy factorization 48 of S In Subsection 5 we give some formulation of braids and regeneration rules In Subsection 5 we compute 48 and in Subsection 5 we give some of its properties 5 The regeneration rules Recall that Z ij is a counterclockwise halftwist of two points i and j We start with an illustrated example of conjugated braids Example 6 The path on the left-hand side in Figure 6 is constructed as follows: take a path z 4 and conjugate it by the fulltwist Z ( encircles counterclockwise while moving above 0 0 Figure 5 Second step of regeneration

10 the axis) We get the left-hand side path z Z 4 The right-hand side one is constructed as follows: take again z 4 and conjugate it first by Z ( encircles counterclockwise) and then by Z ( encircles counterclockwise while moving below the axis) We get the right-hand side path z Z Z 4 The related halftwists of z Z 4 and z Z Z 4 are Z Z 4 and Z Z Z 4 respectively 4 4 Figure 6 Example of conjugated braids In Section 4 we explained that by the regeneration one gets the original branch curve S Let N, M, u be as in Section The set K = S C u is the intersection points of the curve S with the typical fiber C u ; #K = 48 Recall that by the regeneration, each point c in K 0 = S 0 C u is replaced by two close points in K, say c, c We define the braid Zij to be a fulltwist of j around i, and Zi j to be a fulltwist of j around i The braid Zii,j is obtained by a regeneration (the point i on the typical fiber is replaced by i, i ) and it is a fulltwist of j around i and i This braid (and similar ones) is formulated in the following lemma: Lemma 7 [, Lemma 5] The following formulas hold: Zii,j = Z i j Z ij, Z i,jj = Z i j Z i j, Z i,jj = Z i j Z Z i j Z i j, Z ii,j = Z ij Z i j, Z ii,jj = Z i,jj Z i,jj, Z ii,jj = Z i,jj Z i,jj i j, Z i,jj = In the following regeneration rules we shall describe what happens to a factor in a factorized expression of a braid monodromy by a regeneration Theorem 8 First regeneration rule [6, p 6] A factor of the form Z ij regenerates to Z ij Z i j Theorem 9 Second regeneration rule [6, p 7] A factor of the form Zij regenerates to Z ii,j, Z i,jj or Z ii,jj Theorem 0 Third regeneration rule [6, p 7] A factor of the form Zij 4 regenerates to Zi,jj = (Z ij) Z jj (Zij) (Zij) Z jj or to Zii,j = (Zij )Z ii (Zij ) (Z ij )Z ii

11 5 The factorization The braid monodromy factorization 48 of S is 0 C i ϕ i, where C i are the regenerations of the parasitic intersection braids from Section 4, and ϕ i are the local braid monodromies which we get when regenerating around the singularities,,0 i= 5 Braid monodromies related to -points The -points are,, 5, 7 We concentrate in the neighbourhood of, see Figure First the diagonal line 9 regenerates to a conic Q 9 which is tangent to the two other lines and 9, see Lemma 5 We compute the braid monodromy ϕ of the resulting curve Proposition The local braid monodromy ϕ is ϕ = Z Z 9Z 9 9 Z 4 9 Z 9 9 Z 9 9 Z 9 Proof We follow Figure Let π : E D E be the projection to E Let {j} 4 j= be singular points of π as follows:, are the tangent points of Q 9 with the lines L 9, L respectively, is the intersection point of the lines L, L 9, and 4 is the branch point in Q 9 Let N = {x(j) = x j j 4}, such that N E E, N E Take u E, such that C u is a typical fiber and M C u is a real point Recall that K = K(M) K = {, 9, 9, 9 }, such that the points are real and < 9 < 9 < 9 Let i = L i K for i =, 9 and {9, 9 } = Q 9 K We are looking for ϕ M (δ j ) for j =,, 4 So we choose a g-base {δ j } 4 j= of π (E N, u), such that each δ j is constructed from a path γ j below the real line and a counterclockwise small circle around the points in N The diffeomorphism which is induced from passing through a branch point was defined in [] and [] by I R < k > We recall the precise definition from [] Consider a typical fiber on the left side of a branch point (locally defined by y x = 0) The typical fiber intersects the conic in two complex points Passing through this point, these points become real on the right-hand side typical fiber The two points move to the k th place and rotate in a counterclockwise 90 o twist They become real and numbered as k, k + Find first the skeleton ξ x j related to each singular point, as explained in Section Then compute the local diffeomorphisms δ j induced from singular points j Since the points

12 and (resp ) are tangent points (resp a node), the diffeomorphisms δ and δ are each of degree (resp ) j ξ x j ǫ j δ j < 9, 9 > 4 < 9, 9 > <, 9 > <, 9 > < 9, 9 > 4 < 9, 9 > 4 < 9, 9 > I R < 9 > Using [, Theorems 4, 44] and [], we compute the skeleton (ξ x j )Ψ γ j applying to the skeleton ξ x j the product δ i i=j to each j by (ξ x )Ψ γ =< 9, 9 >= z 9 9 ϕ M (δ ) = Z (ξ x )Ψ γ =<, 9 > < 9, 9 >= z Z ϕ M (δ ) = Z Z (ξ x )Ψ γ =< 9, 9 > <, 9 > < 9, 9 >= z 9 ϕ M (δ ) = Z (ξ x 4 )Ψ γ 4 =< 9, 9 > < 9, 9 > <, 9 > < 9, 9 >= z Z 9 9 Z ϕ M (δ 4 ) = Z 9 9 Z 9 9 Z In the second step of the regeneration, each tangent point regenerates to three cusps and we have the following: Proposition The monodromy ϕ regenerates to ϕ = Z 9 9,9 (Z 9 Z 9 Z 9 Z 9 Z 9 Z 9 Z 9 Z 9 )Z 9 9,9 Z,9 Z 9 9 Z 9 9,9 Z,9

13 Proof By Theorem 0, in the regeneration process each one of the tangent points regenerates to three cusps Therefore, the factors Z and Z4 9 regenerate to Z 9 9,9 and Z,9 respectively As explained in Subsection 4, the node is replaced by eight branch points; Since the lines and 9 double to be, and 9, 9 respectively, we get four nodes, the intersections of and 9, and 9, and 9, and 9 Each node regenerates to two branch points For and 9 we get easily the braid monodromy Z 9 Z 9 For the other ones we get Z 9 Z 9, Z 9 Z 9 to the braids in ϕ and Z 9 Z 9 In Figure 7 we show the paths which are related Figure 7 The computations for the other points are similar Therefore we have Proposition The local braid monodromies, induced from the regeneration around the points, 5 and 7 are ϕ = Z4 4,4 (Z 4 Z 4 Z 4 Z 4 Z 4 Z 4 Z 4 Z 4 )4 4,4 Z,4 Z 4 4 Z 4 4,4 Z,4, ϕ 5 = Z7 7, (Z 5 7 Z 5 7 Z 5 7 Z 5 7 Z 5 7 Z 5 7 Z 5 7 Z 5 7 )7 7, Z5 5, Z Z 7 7, Z 5 5,, ϕ 7 = Z,7 (Z 8 Z 8 Z 8 Z 8 Z 8 Z 8 Z 8 Z 8 ),7 Z8 8,7 Z 7 7 Z,7 Z 8 8,7 5 Braid monodromies related to 6-points The 6-points in S 0 are, 4, 6, 8, 9, 0 Regenerations in neighbourhoods of 6-points were studied carefully in [], [7] and [6] In Section 4 we recalled the regeneration around the 6-point Regenerations around 4, 6, 8, 9, 0 are different from the one of only by a modification of indices

14 Since braid monodromies related to regenerations of 6-points were computed in many works (such as [], [6], [7], [9]), we just state our resulting monodromies Theorem 4 The local braid monodromy ϕ has the following form ϕ =Z, ( Z, ii ) Z,0 0 ( (Z, ii )Z, ) Z Z,0 0 Z, ( (Z 0, ii )Z, ) where i=,0, i=,0, i=,0 Z0, Z Z,0, ( (Zii,0 )Z ii,0 Z 0, Z, ) Z Z,0 Z 0, Z, 0 0 i=,0 ) Z (F F Z Z 0 0, Z, Z 0, Z 0,0 0, F = Z, Z 0 (Z 0 )Z, (Z,0 )Z, (Z Z 0 Z Z Z 0 Z Z ) The paths related to the factors in ϕ are Figure Figure Figure 0

15 Figure Figure Figure Figure Figure Figure Figure Figure 8

16 Figure Figure Figure Figure Figure Figure Figure 5

17 0 0 0 Figure Figure 7 In the following theorem we give the local braid monodromies ϕ 4, ϕ 6, ϕ 8, ϕ 9 and ϕ 0 Related figures can be easily constructed by following the ones in Theorem 4 Theorem 5 () The local braid monodromy ϕ 4 has the following form ϕ 4 =Z,4 4 Z, ZZ, Z,4 4 ( Z 6,5 5 Z 5 5,6 ( where i=4, i=5,6,5 (Z6,ii )Z,4 4 ) ( Z,ii ) ( i=4, i=5,6,5 Z,ii ) ZZ 5 5,6 Z 6, ) Z (Z6,ii )Z,4 4 ) (F F Z 4 4 Z 5 5 6,5 5 Z 6, Z,4 4, F = Z 4 5 Z 5,5 5 (Z 4 5 )Z 5,5 5 Z 4 5 (Z 4,5 5 )Z 4 5 ( Z Z 5 5 Z 4 5 Z Z Z 5 5 Z 5 5 Z 4 5 Z ) The paths related to Z Z 5 5 Z 4 5 Z Z Z 5 5 Z 5 5 Z 4 5 Z appear in Figure Figure 8

18 () The local braid monodromy ϕ 6 has the following form ϕ 6 =Z 6,7 7 ( i=8,, Z, Z 8 8, ( (Z 6, ii )Z 6,7 7 ) Z 6,6 6 ( i=7,6 ) Z (F F Z 6 6 Z 8 8, Z 7 7, Z 6,7 7, i=8,, Z 6,ii ) ZZ 6,6 6 Z 6, ( (Z,ii )Z ii, Z 6,7 7 ) Z Z 8 8, Z, Z,6 6 i=7,6 (Z,ii )Z 6,7 7 ) ϕ 8 =( where F = Z 8 8,6 Z 6 (Z 6 )Z 8 8,6 (Z 8 8, )Z 8 8,6 (Z Z 8 Z Z Z 8 Z 8 6 Z ) Moreover, the paths related to Z Z 8 Z Z Z 8 Z 8 6 Z are the same ones as in Figure, with exchange of indices 7, 8, 6, 0 () The local braid monodromy ϕ 8 has the following form i=,,4 ( i=9,8 Z 0,ii ) Z 9 9,0 Z 0,8 8 ( Z,ii ) Z, ( i=9,8 ) Z (F F Z 9 9 Z 9 9, Z 9 9,0 Z,, i=,,4 (Z 0,ii )Z ii,8 8 ) Z Z 9 9,0 Z 0,8 8 Z 0, 0 0 (Z,ii )Z, Z ii, Z 9 9,0 ) Z,4 4 ZZ,4 4 Z, where F =Z 9 Z,8 8 (Z 9 )Z,8 8 Z 9 (Z 9,8 8 )Z 9 ( Z Z 8 Z 9 8 Z Z Z 8 8 Z 8 Z 9 8 Z ) Moreover, the paths related to Z Z 8 Z 9 8 Z Z Z 8 8 Z 8 Z 9 8 Z are the same ones as in Figure 8, with exchange of indices 5 8, 4, 4 9, 5 (4) The local braid monodromy ϕ 9 has the following form ϕ 9 =Z 4,5 5 Z,4 Z Z,4 Z 4, ( i=6,7,8 Z 7,8 8 Z 6 6,7 Z Z 6 6,7 Z 7, ( ) Z (F F Z Z 6 6,4 Z 7,8 8, i=,5 Z 4,ii ) ( i=6,7,8 (Z ii,7) Z,4 ) ( Z 4,ii ) i=,5 (Z ii,7 )Z,4 )

19 where F = Z,8 8 Z 5 5,6 (Z Z 5 5 Z 5 Z Z Z 5 5 Z 5 Z 6 8 Z ) Z Z 6,5 5 Z 6 Moreover, the paths related to (Z Z 5 5 Z 5 Z Z Z 5 5 Z 5 Z ) appear in Figure Figure Z (5) The local braid monodromy ϕ 0 has the following form ϕ 0 =Z 9,0 0 Z 9,4 4 ZZ 9,4 4 Z 9, ( Z, ZZ, Z, Z, ( ) Z (F F Z 0 0 Z, Z 0 0, Z 9,0 0, i=,, i=0,4 Z 9,ii ) ( i=,, (Z,ii )Z 9,0 0 ) ( Z 9,ii ) i=0,4 (Z,ii )Z 9,0 0 ) where F =Z 0 Z,4 4 (Z 0 )Z 0, Z 0, (Z Z Z 0 Z 4 4 Z Z Z 0 Z 4 Z ) Moreover, the paths related to (Z Z Z 0 Z 4 4 Z Z Z 0 Z 4 Z ) are the same ones as in Figure 9, with exchange of indices 0, 5, 6, 8 4

20 5 The products C i The regeneration regenerates the parasitic intersection braids { C i } 0 i=, see Theorem 9 By [6, Lemma 9], we can take the complex conjugations ()( ) D =D = D = Id, D 4 = Z,4 4, D 5 = D 7 = D = D 4 = D 7 = D 0 = D = 4 i= 5 i= i= i,4 i= i 8 8 i=4 i 0, 8 (6)(6 ) Z ii,7 7, D 8 = 5 Zii,8 8, D 9 = i= (9) (0 ) Z ii,, D = ()( ) Z ii,4 4, D 5 = () (6 ) Z ii,7 7, D 8 = i= i 6 8,,6 7 i= (9)(9 ) Z ii,0 0, D = i= i 6 8 i= Zii,5 5, D 6 = i= 8 Zii,9 9, D 0 = i= (9) ( ) Z ii,, D = i=4 i 0 () (4 ) Z ii,5 5, D 6 = () (7 ) Z ii,8 8, D 9 = 8 i= i 6,5 (9) ( ) Z ii,, D 4 = 7 i= i 9 Zii,6 6, i= 8 i=4 Z ii,, i= i 6 8, 8 i= i 9 (9) (0 ) Z ii,, D = (9) ( ) Z ii,4 4 Z ii,9 9, 8 i= i 5,7 (9)(9 ) Z ii,0 0, () (5 ) Z ii,6 6, (9) ( ) Z ii,, Let us denote the regenerations of C i as C i, i 0 These ones are the factorizations of the suitable D t, as in () 5 Properties of the factorization 48 The braid monodromy factorization 48 product 0 C i ϕ i We verify that there are no missing braids i= is the Proposition 6 The braid monodromy factorization of S is 48 Proof By Proposition V in [], deg 48 = = 56 Now we check the degree of the factorization 0 C i ϕ i The monodromies {ϕ i } i=,,5,7 each consist of six cusps and nine branch points, see Section 5 We combine their degrees to get 4 (6 + 9) = 4 7 = 08 i=

21 In each ϕ i, i =, 4, 6, 8, 9, 0 (Section 5), deg F(F) = 48 The factors outside deg F(F) are 0 degree two factors, degree three factors, degree one factors Combining the degrees of these, we get = 78 Thus deg ϕ i = = 6 for i =, 4, 6, 8, 9, 0, and therefore deg( ϕ i ) = 6 6 = 756 i=,4,6,8,9,0 The products {C i } 0 i= consist of 696 nodes, see Subsection 5 We combine degrees of their factors to get deg 0 C i = 696 = 9 i= i= Finally deg 0 C i ϕ i = = 56 Therefore 48 is the desired braid monodromy factorization Now we study invariance properties of 48 Invariance properties are results in which we prove that the braid monodromy factorization 48 is invariant under certain elements of B 48 Establishing invariance properties is essential in order to simplify the computations which follow from the van Kampen Theorem (see [4]) In the end of this section we describe the effect of these properties on the application of the van Kampen Theorem The following definitions are necessary in order to prove the Invariance Properties Definition 7 Let g g k = h h k be two factorized expressions of the same element in a group G We say that g g k is obtained from h h k by a Hurwitz move if p k, such that g i = h i (i p, p + ) g p = h p h p+ h p and g p+ = h p or g p = h p+ and g p+ = h p+h p h p+ In general, g g k h h k (Hurwitz equivalent) if g g k is obtained from h h k by a finite number of Hurwitz moves Definition 8 Let g g k be a factorized expression in G and h G We say that g g k is invariant under h if it is Hurwitz equivalent to (g ) h (g k ) h, where (g i ) h = h g i h Invariance properties are important in view of Lemma VI4 in []: If a braid monodromy factorization n is invariant under h then the equivalent factorization ( n ) h is also a braid monodromy factorization Now we quote invariance rules (conclusions of Theorems 8, 9, 0), we prove Chakiri s Lemma and state some invariance remarks Invariance Rules () A braid Zij is invariant under (Z ii Z jj )q for any q Z

22 () A braid Zi,jj (resp Z ii,jj ) is invariant under Zq jj (resp Z p ii Z q jj ) for any p, q Z () Zi,jj is invariant under Zq jj for any q Z Lemma 9 Chakiri Let g = g g k be a factorized expression in a group G Then g g k is invariant under g m for any m Z Proof It is enough to prove it for a free group G and free generators g g k and for m = Let h be the inner automorphism in G defined by conjugation by g Then (g ) g (g k ) g = (g g k ) g = (g) g = g = g g k Since h(g i ) = (g i ) g, {(g i ) g } are also free generators, and we can use Artin Theorem I4 of [] to conclude that (g ) g (g k ) g is equivalent to g g k Invariance Remarks () To prove invariance of g g k under h it is enough to prove that g g t and g t+ g k are invariant under h Thus we can divide a factorization into subfactorizations and prove invariance on each part separately () An element g is invariant under h if and only if g commutes with h () If a product of elements that commutes with h is invariant under h, the corresponding factorizations are equal (4) If two paths σ and σ do not intersect, the corresponding halftwists H(σ ) and H(σ ) commute (5) If g is invariant under h and h then g is invariant under h h We finally prove the Invariance Properties of 48 Denote Z ij = H(z ij ) Remark 0 The halftwists Z ii and Z jj commute for all i, j, since the path from i to i does not intersect the path from j to j Lemma Each monodromy among ϕ, ϕ, ϕ 5, ϕ 7 is invariant under 4 j= Z m j jj for m j Z Proof The braids in ϕ, ϕ, ϕ 5, ϕ 7, arise from cusps and branch points (Propositions and ) By Invariance Rule, each braid of the form Z ii,j is invariant under Zq ii Each of the braids of the form Z ij is invariant under (Z ii Z jj )q, and in particular, if the braid is of the form Z ii, then it is invariant under Zq ii By Remark 0, each Z kk commutes with the above braids when k i, j

23 Therefore, each one of the monodromies ϕ, ϕ, ϕ 5, ϕ 7 is invariant under 4 j= Z m j jj for m j Z Lemma Each factorization {C i }, i =,,0, is invariant under 4 j= Z m j jj for m j Z Proof We apply Invariance Rule and Invariance Remark 4 on each C i to get the desired invariance Now we prove invariance for the monodromy ϕ A similar proof will hold for each one of the monodromies ϕ 4, ϕ 6, ϕ 8, ϕ 9, ϕ 0 Lemma ϕ is invariant under (Z Z 0 0 ) p (Z Z 0 0 ) q (Z Z ) r for all p, q, r Z Proof Case : p = q = r As proved in [6, Lemma ], the monodromy ϕ can be written as Lemma 9, ϕ is invariant under ( j=,,,0,,0 is invariant under (Z Z 0 0 ) p (Z Z 0 0 ) p (Z Z ) p Case : p = 0 Denote: ǫ = (Z Z 0 0 ) q (Z Z ) r () Step : Factors outside of F F Z Z 0 0 Z Z,0 0 Z, and Z Z,0 Z 0, Z, 0 0 commute with ǫ j=,,,0,,0 Z jj By Z jj ) p Since is a central element, ϕ Z,, Z,0 0, Z 0, and Z,0Z, are invariant under ǫ by Invariance Rule The degree factors are of the form Z αα,β where β =,, 0, 0 By Invariance Rule, they are invariant under Z αα, and since the other halftwists in ǫ commute with Z αβ and Z α β, we get that Z αα,β is invariant under ǫ Moreover all conjugations which appear in ϕ (ie ( ) ) are invariant under ǫ by Invariance Rule and by Invariance Remark 4 () Step : Factors in F F Z Z 0 0 Denote ρ = Z Z 0 0 In order to prove that F F ρ is invariant under ǫ, we consider the following subcases: Subcase : q = 0 and ǫ = (Z Z ) r Z, and (Z,0 )Z, are invariant under Z (Invariance Rule ) and commute with Z (Invariance Remark 4) Thus Z, (Z,0 )Z, is invariant

24 under ǫ (Invariance Remarks and 5) Z 0 and (Z 0 )Z, commute with Z and Z and thus with ǫ Z Z 0 Z Z Z 0 Z Z is invariant under (Z Z ) r by Invariance Rule For F ρ we have the same conclusion, since Z Z 0 0 commutes with ǫ = (Z Z ) r Subcase : r = 0, q = and ǫ = Z Z 0 0 Notice that here ǫ = ρ To prove that F F ρ is invariant under ρ, it is necessary to prove that F F ρ is Hurwitz equivalent to F ρ F Since AB is Hurwitz equivalent to BA B, it is enough to prove that F F ρ is Hurwitz equivalent to F (F ρ ) F Thus it is enough to prove that F ρ is Hurwitz equivalent to (F ρ ) F or that (F ρ ) F is Hurwitz equivalent to F ρ By Theorem 4, F = 8ρ Z Z (F ) ρ, thus F = F ρ (Z Z ) ρ 8 Now we have (F ρ ) F = (F ρ ) F ρ (Z Z ) ρ 8 = as factorized expression (F ρ ) F ρ (Z Z ) ρ Chakiri (F ρ ) (Z Z ) ρ = (F (Z Z ) ) ρ F ρ Subcase Subcase : q = q ; ǫ = (Z Z 0 0 ) q (Z Z ) r F F ρ can be written as 8 (Z Z 0 0 ) (Z Z ) By Chakiri s Lemma, F F ρ is invariant under ( 8 (Z Z 0 0 ) (Z Z ) ) q and thus under (Z Z 0 0 ) q (Z Z ) q By Subcase, F F ρ is invariant under (Z Z ) r q By Invariance Remark 5, F F ρ is invariant under (Z Z 0 0 ) q (Z Z ) q (Z Z ) r q = ǫ Subcase 4: q = q + ; ǫ = (Z Z 0 0 ) q + (Z Z ) r It is easy to prove it by cases, and Invariance Remark 5 Case : p, q, r arbitrary; ǫ = (Z Z 0 0 ) p (Z Z 0 0 ) q (Z Z ) r By case, ϕ is invariant under (Z Z 0 0 ) p (Z Z 0 0 ) p (Z Z ) p, and by case under (Z Z 0 0 ) q p (Z Z ) r p By Invariance Remark 5, ϕ is invariant under ǫ Corollary 4 Each one of the monodromies among ϕ 4, ϕ 6, ϕ 8, ϕ 9, ϕ 0 is invariant under (Z ii Z jj ) p (Z kk Z ll ) q (Z mm Z nn ) r for all p, q, r Z, where i and j are the diagonal lines around

25 the relevant 6-point, while k and l (resp m and n) are the vertical (resp horizontal) ones (see Figure ) Since the proof of the invariance for the 6-points relies on the invariance of each local braid monodromy under such expressions as in Corollary 4, we have to pay attention that each diagonal line (except the lines 4, 7, 9, in Figure ) at a certain 6-point is also a diagonal line at another 6-point For example, the monodromy ϕ is invariant under (Z Z 0 0 ) p (Z Z 0 0 ) q (Z Z ) r for all p, q, r Z (see Lemma ) And the monodromy ϕ 4 is invariant under (Z Z 66 ) p (Z 44 Z 55 ) q (Z 5 5 Z ) r for all p, q, r Z (see Corollary 4) It means that p = p In the same way, ϕ 6 is invariant under (Z 66 Z ) p (Z 6 6 Z ) q (Z 77 Z 88 ) r for all p, q, r Z That gives p = p In the notation of Lemmas and, p = m = m 0, p = m = m 6, p = m 6 = m, therefore m = m 6 = m 0 = m Now, the lines 4 and 7 (resp 9 and ) are diagonal lines at only one 6-point 9 (resp 0) Therefore in the above notations, m 4 = m 7 (resp m 9 = m ) We consider now the vertical lines By a similar argument as above, we can conclude that m = m 6 = m 0 = m from the invariance of ϕ, ϕ 6, ϕ 9 and ϕ 0 Since the lines 4 and 5 (resp 9 and ) are vertical lines at only one 6-point 4 (resp 8), we get m 4 = m 5 (resp m 9 = m ) For the horizontal lines, we can immediately conclude that m = m, m 7 = m 8, and m 5 = m 8 = m = m 4 Corollary 5 The braid monodromy factorization 48 is invariant under (Z Z ) m (Z 44 Z 55 ) m 4 (Z 77 Z 88 ) m 7 (Z 99 Z ) m 9 (Z 4 4 Z 7 7 ) m 4 (Z 9 9 Z ) m 9 (Z Z 66 Z 0 0 Z ) m (Z Z 6 6 Z 0 0 Z ) m (Z 5 5 Z 8 8 Z Z 4 4 ) m 5 By the above corollary we can do the following Let z ij be a path connecting i or i with j or j and Z ij its corresponding halftwist We can conjugate Z ij by Z ± ii, Z ± jj These conjugations are the actions of Z ± ii (resp Z ± jj ) on the head (resp tail ) of z ij within a small circle around i and i (resp j and j ) The body of z ij does not change under such conjugations and in particular not under Z ±m ii or Z ±n jj for m, n Z

26 We are interested in the fundamental group of the Galois cover of the surface, and in particular in π (CP S) In [4] we will find from the factorization 48, via the van Kampen Theorem, a presentation for π (CP S) Corollary 5 enables us to use Theorem 6 from [8] (given a factorization of a curve, such that a sub-factorization of it is invariant under some element h and induces a relation Γ i Γ it on π (CP S), then (Γ i ) h (Γ it ) h is also a relation) That means that we can expand our list of relations for the convenience of the computations in [4] 6 Acknowledgements This research was initiated while the first authors was staying at the Mathematics Institute, Erlangen - Nürnberg university, Germany She wishes to thank the institute for the hospitality and to the hosts Wolf Barth and Herbert Lange She wishes to thank also the Einstein Institute for Mathematics (Jerusalem) for her present stay, and to the hosts Hershel Farkas and Ruth Lawrence-Neumark References [] Artin, E, On the fundamental group of an algebraic curve, Ann Math 48, 947, 0-0 [] Artin, E, Theory of braids, Ann Math 48, 947, 0-6 [] Amram, M, Galois Covers of Algebraic Surfaces, PhD Dissertation, 00 [4] Amram, M, Ciliberto, C, Miranda, R, Teicher, M, Vishne, U, Fundamental group of a Galois cover of a non-prime K surface, submitted [5] M Amram, D Garber and M Teicher, Fundamental groups of tangented conic-line arrangements with singularities up to order 6, submitted [6] Amram, M, Goldberg, D, Teicher, M, Vishne, U, The fundamental group of the Galois cover of the surface CP T, Algebraic and Geometric Topology, Volume, 00, no 0, 40-4 [7] Amram, M, Teicher, M, On the degeneration, regeneration and braid monodromy of T T, Acta Applicandae mathematicae, 75(), 00, [8] Amram, M, Teicher, M, The fundamental group of the complement of the branch curve of the surface T T in C, Osaka, Japan, 40(4), 00 [9] Amram, M, Teicher, M, Vishne, U, The fundamental group of the Galois cover for Hirzebruch surface F (, ), submitted [0] Amram, M, Teicher, M, Vishne, U, The Coxeter quotient of the fundamental group of a Galois cover of T T, to appear

27 [] Amram, M, Teicher, M, Vishne, U, The fundamental group of the Galois cover of the surface T T, submitted [] Auroux, D, Donaldson, S, Katzarkov, L, Yotov, M, Fundamental groups of complements of plane curves and symplectic invariants, Topology 4, 004, 85-8 [] Chisini, O, Courbes de diramation des planes multiple et tresses algebriques, Deuxieme Colloque de Geometrie Algebrique tenu a Liege, CBRM, 95, -7 [4] Ciliberto, C, Lopez, A, Miranda, R, Projective Degenerations of K Surfaces, Gaussian Maps, and Fano Threefolds Inventiones mathematicae, vol 4, 99, [5] Ciliberto, C, Miranda, R, Teicher, M, Pillow degenerations of K surfaces, Applications of Algebraic Geometry to Coding Theory, Physics and Computations, NATO Science Series II/6, Kluwer Acad Publish, 00, 5-64 [6] Kulikov, V, Degenerations of K Surfaces and Enriques Surfaces Math USSR Izvestija, 977, [7] Moishezon, B, Algebraic surfaces and the arithmetic of braids, I, Arithmetic and Geometry, papers dedicated to IR Shafarevich, Birkhäuser, 98, [8] Moishezon, B, Robb, A, Teicher, M, On Galois covers of Hirzebruch surfaces, Math Ann 05, 996, [9] Moishezon, B, Teicher, M, Existence of simply connected algebraic surfaces of positive and zero indices, Proceedings of the National Academy of Sciences, 8, 986, [0] Moishezon, B, Teicher, M, Galois coverings in the theory of algebraic surfaces, Proc of Symp in Pure Math 46, 987, [] Moishezon, B, Teicher, M, Simply connected algebraic surfaces of positive index, Invent Math 89, 987, [] Moishezon, B, Teicher, M, Braid group technique in complex geometry I, Line arrangements in CP, Contemporary Math 78, 988, [] Moishezon, B, Teicher, M, Braid group technique in complex geometry II, From arrangements of lines and conics to cuspidal curves, Algebraic Geometry, Lect Notes in Math Vol 479, 990 [4] Moishezon, B, Teicher, M, Finite fundamental groups, free over Z/cZ, Galois covers of CP, Math Ann 9, 99, [5] Moishezon, B, Teicher, M, Braid group technique in complex geometry III: Projective degeneration of V, Contemp Math 6, 99, - [6] Moishezon, B, Teicher, M, Braid group technique in complex geometry IV: Braid monodromy of the branch curve S of V CP and application to π (CP S, ), Contemporary Math 6, 99, -58

28 [7] Moishezon, B, Teicher, M, Braid group technique in complex geometry V: The fundamental group of a complement of a branch curve of a Veronese generic projection, Communications in Analysis and Geometry 4(), 996, -0 [8] Moishezon, B, Teicher, M, Fundamental groups of complements of branch curves as solvable groups, Israel Mathematics Conference Proceedings (AMS Publications), vol 9, 996, 9-46 [9] Moishezon, B, Teicher, M, Braid groups, singularities and algebraic surfaces, Academic Press, to be published Birkhäuser [0] Robb, A, On branch curves of algebraic surfaces, Singularities and Complex Geometry AMS/IP Stud Adv Math 5, 997, 9- [] van Kampen, ER, On the fundamental group of an algebraic curve, Amer J Math 55, 9, Meirav Amram, Mathematisches Institut, Bismarck Strasse /, Erlangen, Germany; Einstein Institute for Mathematics, Hebrew University, Jerusalem address: meirav@macsbiuacil/ameirav@mathhujiacil Ciro Ciliberto, Dipartimento di Matematica, Universita di Roma II, Tor Vergata, 00 Roma, Italy address: cilibert@matuniromait Rick Miranda, Department of Mathematics, Colorado State University, Fort Collins, CO 805 USA address: rickmiranda@colostateedu Mina Teicher, department of mathematics, Bar-Ilan university, 5900 Ramat-Gan, Israel address: teicher@macsbiuacil