NEW INVARIANTS FOR SURFACES

Mina Teicher

Abstract. We define a new invariant of surfaces, stable on connected components of moduli space. The new invariant comes from the polycyclic structure of the fundamental group of the complement of a branch curve. We compute this invariant for a few examples. Moreover, we indicate the possibility of using braid monodromy factorizations of branch curves as an invariant of a surface. (Recall that braid monodromy factorizations related to curves is a first step in computing the fundamental group of the complement of the curve.)

0. Introduction

After the remarkable discovery of the Donaldson invariants and the Sieberg-

Witten invariants [SW], there was hope that using them one could distinguish

between different connected components of moduli space of surfaces. In our earlier

work we indicated that we believe that these invariants are not fine enough for

this differentiation and a more geometrical approach is needed. Indeed, in 1997,

M. Manetti produced examples of surfaces which are diffeomorphic but are not a

deformation of each other; and thus it is clear that a more direct geometric approach

is needed. We went on to suggest the following distinguishing invariant:

Let X be a complex algebraic surface of general type embedded in ℂP^{n}. Take a

generic projection of X to ℂP^{2} and let S_{X} be its branch curve. Clearly π_{1}(ℂP^{2} −

S_{X}) is stable on a connected component of moduli space of surfaces. We believe

1991 Mathematics Subject Classification. 20F36, 14J10. Key words and phrases. Algebraic surfaces, branch curves, fundamental group, braid group,

braid monodromy factorization.

The author wishes to thank the Max-Planck Institute for a fruitful and pleasant stay while writing this paper. This paper was partially supported by the Emmy Noether Research Institute for Mathematics, Bar-Ilan University, and the Minerva Foundation of Germany.

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that these groups can distinguish between different components. We base our belief on the structure of such groups which already have been computed.

1. History of computations of fundamental groups of complements of branch curves

Let X be a complex algebraic surface of general type embedded in ℂP^{N}. Let f : X → ℂP^{2} be a generic projection and let S_{X} ∈ ℂP^{2} be its branch curve. Let ℂ^{2} be a big affine piece of ℂP^{2} s.t. S_{X} is transversal to the line at infinity. We denote:

Ḡ_{X} = π_{1}(ℂP^{2} − S_{x}), G = π_{1}(ℂ^{2} − S_{X}).

The first computations of Ḡ_{X} were done by Zariski [Z]. He computed Ḡ_{X} for X a cubic surface in ℂP^{2}, to be ℤ_{2}∗ℤ_{3} (free products of 2 finite cyclic groups). The topic was renewed by Moishezon in the late 1970’s, when he generalized Zariski’s result to X_{n}, a deg n surface in ℂP^{3}, and proved that in this case G_{X_{n}} is the braid group B_{n} and Ḡ_{n} is the braid group over its centers, B_{n}/center. In fact, Moishezon’s result [Mo1] for n = 3 coincides with Zariski’s result since B_{3}/center ≃ ℤ_{2} ∗ ℤ_{3}. (B_{3} = ⟨x, y | xyx = yxy⟩, center B_{3} = ⟨(xy)^{3}⟩. Thus in B_{3}/center, x = (xy)^{3}x = xy ·yx·xyx and y = y(xy)^{3} = yxy ·(xy)^{2} = xyx·(xy)^{2}. Thus B_{3}/center is generated by xy and xyx while (xy)^{3} = 1 and (xyx)^{2} = xyx · yxy = (xy)^{3} = 1).

The next example was V_{2} Veronese of order 2 [MoTe3]. In all the above examples, G contained a free noncommutative subgroup of 2 elements (and in other related examples as in [DOZ]). We call such group “big”. This gave the basis to the feeling that this will always be the case. Since 1991 new examples were computed and these examples were not “big”.

NEW INVARIANTS FOR SURFACES 3

2. The new examples: almost solvable groups

To our great surprise the invariants that were computed after 1991 were not “big”. It turns out that in all of the new examples, Ḡ_{X} = P ψ_{1}(ℂP^{2} − S_{X}) and G_{X} = π_{1}(ℂ^{2} − S_{X}) satisfy the following conditions:

1) There exists a quotient of the braid group namely B̃_{n} s.t. B̃_{n} acts on G_{X} and Ḡ_{X}.

2) G_{X} and Ḡ_{X} are not “big”.

3) G_{X} and Ḡ_{X} are “small”: They are almost solvable (or virtually solvable in another terminology), i.e., they contain an almost solvable subgroup of finite index.

4) Moreover, G_{X} and Ḡ_{X} are extensions of a solvable group by a symmetric group.

5) Ḡ_{X} = G/central element.

Moreover, in all the new examples we have the following series:

(1)  {1} ◁ A_{0} ◁ A_{1} ◁ A_{2} ◁ A_{3} ◁ G_{x}

where G/A_{3} ≃ S_{n} and A_{i}/A_{i−1} is a direct sum of ℤ and one or two finite cyclic groups, to some power (ℤ + ℤ_{p} + ℤ_{q})^{t}. For example we have the following computations: 2.1. V_{p} Veronese of order p, p ≥ 3. There exists a series

1 = A_{0} ◁ A_{1} ◁ A_{2} ◁ A_{3} ◁ G_{V_{p}} s.t.

G/A_{3} ≃ S_{p^{2}}, symmetric groups on p^{2} elements.

A_{3}/A_{2} ≃ ℤ

A_{2}/A_{1} ≃ \{ \begin{matrix}(ℤ ⊕ ℤ_{3})^{p^{2}−1} & p ≡ (mod 3) \\ ℤ^{p^{2}−1} & p ≢ 0(mod 3) \end{matrix}

A_{1}/A_{0} ≃ \{ \begin{matrix}ℤ_{2} & p odd \\ 0 & p even \end{matrix}

Moreover, we know that A_{0} is the commutant subgroup of A_{3}, and A_{0} ⊂ Center(G). There exists a series

1Ā_{0} ◁ Ā_{1} ◁ Ā_{2} ◁ Ā_{3} ◁ Ḡ_{p} s.t.

Ḡ_{p}/Ā_{3} ≃ S_{p^{2}}

Ā_{3}/Ā_{2} ≃ ℤ_{q} q = \frac{3p(p − 1)}{2}

Ā_{2}/Ā_{1} ≃ \{ \begin{matrix}(ℤ ⊕ ℤ_{3})^{p^{2}−1} & p ≡ 0(mod 3) \\ ℤ^{p^{2}−1} & p ≢ (mod 3) \end{matrix}

Ā_{1}/Ā_{0} = \{ \begin{matrix}ℤ_{2} & p odd \\ 0 & p even \end{matrix}

2.2. X_{a} projective embedding of ℂP′ × ℂP′ w.r.t. to | al_{1} + bl_{2} |. There exists a series

1 ◁ A_{0} ◁ A_{1} ◁ A_{2} ◁ A_{3} ⊂ G_{X_{ab}} s.t.

for n = 2ab = deg X_{ab} we have:

G_{X_{ab}}/A_{3} ≃ S_{n}
A_{3}/A_{2} ≅ ℤ

A_{2}/A_{1} ≅ \{ \begin{matrix}(ℤ_{2})_{0}^{n−1}(ℤ_{a−b})^{n−1} & a, b even \\ (ℤ_{2(a−b)}^{n−1}) & otherwise \end{matrix}

A_{1}/A_{0} ≃ \{ \begin{matrix}ℤ_{2} ⊕ ℤ_{2} & a, b even \\ ℤ_{2} & otherwise \end{matrix}

There exists a series

1 ◁ Ā_{=} ◁ Ā_{1} ◁ Ā_{2} ◁ Ā_{3} ◁ Ḡ_{X_{ab}}

s.t.

Ḡ_{X_{ab}}/Ā_{3} ≃ S_{n}
Ā_{3}/Ā_{2} ≃ Z_{m_{1}} m_{1} = 3ab − a − b
Ā_{2}/Ā_{1} ≃ A_{2}/A_{1}
Ā_{1}/Ā_{0} ≃ A_{1}/A_{0}

2.3. Complete intersection of deg n (not a hypersurface). There exists a series

1A_{0} ◁ A_{1} ◁ A_{2} ◁ A_{3} ◁ G

s.t.

G/A_{3} ≃ S_{n}
A_{3}/A_{2} ≃ ℤ
A_{2}/A_{1} ≃ ℤ^{n−1}
A_{1}/A_{0} ≃ ℤ_{2}

In this case G is B̃_{n} itself.

There are other computations in progress on Hirzebruch surfaces (following [MoRoTe] and [FRoTe]), on K3-surfaces (following [CiMT]) and on toric.

6 M. TEICHER

The proofs are based on our braid monodromy techniques as presented in detail in [MoTe4], [MoTe6], [MoTe7], [MoTe8], [MoTe9], [MoTe10], [Te2], [Ro], [FRoTe], and on the Van Kampen Theorem [VK].

In fact, all these groups are polycyclic with A_{1} ⊆ Center(G). A_{1} = A_{3} = A_{2}. Group theoretic classification of these groups might answer questions such as the following: Does every almost polycyclic B̃_{n}-group appear as the fundamental group of complements of branch curves? How many non-isomorphic groups of that type appear? (We transfer the question to a classification problem in group theory.)

The interest in fundamental groups is growing in general, see for example, [ToCa], [Si], [BoKa], [L1], [L2], [RoTe], [To].

3. The mysterious quotient of B_{n} that acts on G_{x} and Ḡ_{X}

In this section we bring the definition of the braid group and we distinguish certain elements, called half-twists. Using half-twists we present Artin’s Structure Theorem for the braid group and the natural homomorphism to the symmetric group. We also define transversal half-twists and the quotient of B_{n} called B̃_{n}, and quote the almost solvability theorem for this group.

Definition 3.1. \underline{Braid group B_{n} = B_{n}[D, K]} Let D be a closed disc in ℝ^{2}, K ⊂ D, K finite. Let B be the group of all diffeomorphisms β of D such that β(K) = K, β |_{∂D}= Id_{∂D}. For β_{1}, β_{2} ∈ B, we say that β_{1} is equivalent to β_{2} if β_{1} and β_{2} induce the same automorphism of π_{1}(D − K, u). The quotient of B by this equivalence relation is called the braid group B_{n}[D, K](n = ♯K). The elements of B_{n}[D, K] are called braids.

Definition 3.2. \underline{H(σ) half-twist defined by σ} Let D, K be as above. Let a, b ∈ K, K_{a,b} = K − a − b and σ be a simple path in D−∂D connected a with b s.t. σ∩K = {a, b}. Choose a small regular neighborhood U of σ and an orientation preserving diffeomorphism f : ℝ^{2} → ℂ^{1}(ℂ^{1} is taken with

NEW INVARIANTS FOR SURFACES 7

the usual “complex” orientation) such that f(σ) = [−1, 1], f(U) = {z ∈ ℂ^{1} ∥ z |< 2}. Let α(r), r ≥ 0, be a real smooth monotone function such that α(r) = 1 for r ∈ [0, \frac{3}{2}] and α(r) = 0 for r ≥ 2.

Define a diffeomorphism h : ℂ^{1} → ℂ^{1} as follows. For z ∈ ℂ^{1}, z = re^{iφ}, let h(z) = re^{i(φ+α(r))}. It is clear that on \{z ∈ ℂ^{1} ∥ z |≤ \frac{3}{2}\}, h(z) is the positive rotation by 180^{∘} and that h(z) = Identity on {z ∈ ℂ^{1} ∥ z |≥ 2}, in particular, on ℂ^{1} − f(U). Considering (f ∘ h ∘ f^{−1}) |_{D} (we always take composition from left to right), we get a diffeomorphism of D which interchanges a and b and is the identity on D − U. Thus it defines an element of B_{n}[D, K], called the half-twist defined by σ and denoted H(σ).

Using half-twists we build a set of generators for B_{n}. Definition 3.3. \underline{Frame of B_{n}[D, K]} Let D be a disc in ℝ^{2}. Let K = {a_{1}, …, a_{n}}, K ⊂ D. Let σ_{1}, …, σ_{n−1} be a system of simple paths in D − ∂D such that each σ_{i} connects a_{i} with a_{i+1} and for

i, j ∈ \{1, …, n − 1\}, i < j, σ_{i} ∩ σ_{j} = \{ \begin{matrix}∅ & if | i − j |≥ 2 \\ a_{i+1} & if j = i + 1. \end{matrix} .

Let H_{i} = H(σ_{i}). We call the ordered system of half-twists (H_{1}, …, H_{n−1}) a frame of B_{n}[D, K] defined by (σ_{1}, …, σ_{n−1}), or a frame of B_{n}[D, K] for short. Notation 3.4.

[A, B] = ABA^{−1}B^{−1}
⟨A, B⟩ = ABAB^{−1}A^{−1}B^{−1}
(A)_{B} = B^{−1}AB.

Theorem 3.5. (E. Artin’s braid group presentation). Let {H_{i}} be a frame of B_{n}. Then B_{n} is generated by the half-twists H_{i} and all the relations between H_{1}, …, H_{n−1} follow from

\begin{matrix}[H_{i}, H_{j}] = 1 & if | i − j |> 1, \\ ⟨H_{i}, H_{j}⟩ = 1 & if | i − j |= 1, \end{matrix}

8 M. TEICHER

1 ≤ i, j ≤ n − 1.

Proof. [A] (or [MoTe4], Chapter 5). Theorem 3.6. Let {H_{i}} be a frame of B_{n}. Then

(i) for n ≥ 2, Center B_{n} is isomorphic to Z with a generator Δ_{n}^{2} = (H_{1} · . . . · H_{n−1})^{n}.

(ii) B_{2} ≃ ℤ with a generator H_{i}.

Proof. [MoTe4], Corollary V.2.3. Proposition 3.7. There is a natural homomorphism B_{n} → S_{n} (symmetric group on n elements) defined by H_{i} → (ii + 1)(i iH). Proof. Since the transportations α_{i} = (ii + 1) satisfy the relations from Artin’s theorem (2.5), the above is well defined. Definition 3.8. \underline{P_{n}} The kernel of the above homomorphism is denoted by P_{n}. Remark 3.9. The transpositions α_{i} satisfy a relation that H_{i} do not satisfy, which is α_{i}^{2} = 1. In fact it is true for any transposition. Under the above homomorphism the image of any half-twist is a transposition and thus any square of a half-twist belongs to ker(B_{n} → S_{n}) which is P_{n}. Definition 3.10. \underline{Transversal half-twist, adjacent half-twist, disjoint half-twist} Let σ_{1} and σ_{2} be 2 paths in D with endpoints in K which do not intersect K otherwise (like in 2.2). The half-twists H(σ_{1}) and H(σ_{2}) will be called transversal if σ_{1} and σ_{2} intersect transversally in one point which is not an end point of either of the σ_{i}’s. The half-twists H(σ_{1}) and H(σ_{2}) will be called adjacent if σ_{1} and σ_{2} have one endpoint in common. The half-twists H(σ_{1}) and H(σ_{2}) will be called disjoint if σ_{1} and σ_{2} do not intersect.

NEW INVARIANTS FOR SURFACES 9

Claim 3.11. Disjoint half-twists commute and adjacent half-twists satisfy the triple relation ABA = BAB. Proof of 3.11. By Proposition 2.7 and the fact that every 2 half-twists are conjugated to each other. Definition 3.12. \underline{B̃_{n}} Let Q_{n} be the subgroup of B_{n} normally generated by [X, Y] for X, Y transversal half-twists. B̃_{n} is the quotient of B_{n} modulo Q_{n}.

A basic property of B̃_{n} is the following: Theorem 3.13. B̃_{n} is an almost solvable group; there exist a series

1 ◁ P̃′_{n} ◁ P̃_{n,0} ◁ P̃_{n} ◁ B̃_{n} s.t.

B̃_{n}/P̃_{n} ≃ S_{n}
P̃_{n}/P̃_{n,0} ≃ ℤ
P̃_{n,0}/P̃′_{n} ≃
P̃′_{n} ≃ ℤ_{2}(⊆ Center b̃_{n}).

Proof. [Te1].

4. Applications of Kulikov’s proof of Chisini conjecture V. Kulikov proved in 1988 [Ku] that the so-called Chisini conjecture is true in

many (and in fact for the most important) families of surfaces. Basically he proved that if a curve S in ℂP^{2} is a branch curve of a generic projection to ℂP^{2}, then it is the branch curve of only one generic projection. Thus if we can distinguish different branch curves by fundamental groups of their complements, we will also be able to distinguish between the corresponding surfaces.

5. The new invariant

We conjecture that for many classes of embedded (in ℂP^{N}) surfaces of general type the fundamental group of the complement of the branch curve of a generic

10 M. TEICHER

projection is almost solvable. Moreover, we believe that like in all the new examples there exist:

\begin{matrix}1 = A_{0} ◁ … ◁ A_{m−1} ◁ A_{m} ◁ G \\ 1 = Ā_{0} ◁ … ◁ Ā_{m−1} ◁ Ā_{m} ◁ G \end{matrix}

s.t. G/A_{m} ≃ Ḡ/Ā_{m} ≃ S_{n} n = deg X

A_{i}/A_{i−1} ≃ (ℤ_{t_{i}} ⊕ ℤ_{s_{i}} + ℤ)^{q_{i}}

Ā_{0}/Ā_{i−1} ≃ (ℤ_{t′_{t_{i}}} ⊕ ℤ_{a′_{i}} ⊕ ℤ^{r′_{1}})^{q′_{i}}

Thus we attach to X the invariants

\begin{matrix}n(X) = (t_{1}, s_{1}, v_{1}, q_{1}, …, t_{m}, s_{m}, v_{m}, q_{m}, n) \\ n̄(X) = (t′_{1}, s′_{1}, y′_{1}, q′_{1}, …, t′_{m}, s′_{m}, v′_{m}, q′_{m}, n). \end{matrix}

Clearly, if X and Y are in the same connected component, these invariants coincide for them and if they do not coincide, they are not in the same component.

Examples 5.1.

n(V_{p}) = (2, 4, 0, 1, 3, 1, 1, p^{2} − 1, 1, 1, 1, 1, p^{2})  p odd
p = 0(3)
= (2, 1, 0, 1, 1, 1, 1, p^{2} − 1, 1, 1, 1, 1, p^{2})  p odd
p ≢ 0(3)
= (1, 1, 0, 0, 3, 1, 1, p^{2} − 1, 1, 1, 1, 1, p^{2})  p even
p = 0(3)
= (1, 1, 0, 0, 1, 1, 1, p^{2} − 1, 1, 1, 1, 1, p^{2})  p even
q ≠ 0(3)

NEW INVARIANTS FOR SURFACES 11

n̄(V_{p}) = (2, 4, 0, 1, 3, 1, 1, p^{2} − 1, \frac{3p(p − 1)}{2}, 1, 0, 1, p^{2})  p odd

p = 0(3)

= (2, 1, 0, 1, 1, 1, 1, p^{2} − 1, \frac{3p(p − 1)}{2}, 1, 0, 1, p^{2})  p odd

p ≢ 0(3)

= (1, 1, 0, 0, 3, 1, 1, p^{2} − 1, \frac{3p(p − 1)}{2}, 1, 0, 1, p^{2})  p even

p = 0(3)

= (1, 1, 0, 0, 1, 1, 1, p^{2} − 1, \frac{3p(p − 1)}{2}, 1, 0, 1, p^{2})  p even

q ≠ 0(3)

n(X_{ab}) = (2, 2, 0, 1, 2, a − b, 0, 2ab − 1, 1, 1, 1, 1, 2ab)  a, b even

otherwise

= (2, 1, 0, 1, 2(a − b), 1, 0, 2ab − 1, 1, 1, 1, 1, 2ab)

n̄(X_{ab}) = (2, 2, 0, 1, 2, a − b, 0, 2ab − 1, 3ab − a − b, 1, 0, 1, 2ab)  a, b even

otherwise

= (2, 1, 0, 1, 2(a − b), 1, 0, 2ab − 1, 3ab − a − b, 1, 0, 1, 2ab)

n(CI) = (2, 1, 0, 1, 1, 1, 1, n − 1, 1, 1, 1, 1, n)

On the other hand, n̄(CI) is not completely determined by n.

6. Example: precise computation of G

In this section we give the precise statement of G_{X} for X the Veronese of order 3. ¿From this structure it will be understood how the almost solvability phenomenon

12 M. TEICHER

came up.

Let G = π_{1}(ℂ^{2} − S) for S the branch curve of V_{3} → ℂℙ^{2}.

In order to formulate the theorem we need a few definitions.

Definitions 6.1. \underline{G_{0}(9)} G_{0}(9) is a ℤ_{2} extension of a free group on 8 elements. We take the following model for G_{0}(9): Let G_{0}(9) be generated by {g_{i}}_{i=1 i≠4}^{9} s.t.

[g_{i}, g_{j}] = \{ \begin{matrix}1 & T_{i}, T_{j} are disjoint \\ τ & otherwise \end{matrix}

where τ^{2} = 1, τ ∈ Center G_{0}(9). We take the following action of B̃_{9} on G_{0}(9)

(g_{i})T̃_{k} = \{ \begin{matrix}g_{i}^{−1}τ & k = i \\ g_{i} & T_{i}, T_{k} are disjoint \\ g_{i}g_{k}^{−1} & T_{i}, T_{k} are not orderly adjacent \\ g_{k}g_{i} & otherwise \end{matrix}

\underline{B̃_{n} ⋉ G_{0}(9)} Consider the semidirect product B̃_{9} ⋉ G_{0}(9) w.r.t. the chosen action. \underline{N_{9}} Let c = [T̃_{1}^{2}, T̃_{2}^{2}]. Let ξ_{1} = (T̃_{2}T̃_{1}T̃_{2}^{−1})^{2}T̃_{2}^{−2}. Let N_{9} ◁ B̃_{9} ⋉ G_{0}(9) be normally generated by cτ^{−1} and (g_{1}ξ_{1}^{−1})^{3}. \underline{G_{9}}

Let G_{9} = \frac{B̃_{9} ⋉ G_{0}(9)}{N_{9}}.

NEW INVARIANTS FOR SURFACES 13

\underline{ψ̂_{9}} Let ψ̃_{9} be the homomorphism B̃_{9} → S_{9} induced from the standard homomorphism B_{9} → S_{9} (see 2.7). ψ̃_{9} exists since [X, Y] → 1 under the standard homomorphism. Let ψ̂_{9} : G_{9} → S_{9} be defined by the first coordinate ψ̂_{9}(α, β) = ψ̃_{9}(α). \underline{ψ} The projection V_{3} → ℂ^{2}, of degree 9, induces a standard monodromy homomorphism π_{1}(ℂ^{2} − S, ∗) → S_{9} which we denote by ψ. Theorem 6.2. G ≃ B̃_{9} ⋉ G_{0}(9)/N_{9} s.t. ψ is compatible with ψ_{9}. Proof. [MoTe9], [MoTe10], [Te2].

We also know about G that: \underline{H_{9}, H_{9,0}, H′_{9}, H′_{9,0}} Let Ab : B_{9} → ℤ be the abelianization of B_{9} and B_{9} over its commutator subgroup. Let Ã_{b} : B̃_{9} → ℤ be a homomorphism induced from Ab (which exists since Ab([X, Y]) = 1). Let Âb : G_{9} → ℤ be defined by the first coordinate Âb(α, β) = Ãb(α). Let H_{9} = ker ψ̂_{9}. Let H_{9,0} = ker ψ̂_{9} ∩ ker Âb. Let H′_{9}, H′_{9,0} be the commutant subgroup of H_{9} and H_{9,0}, respectively. Proposition 6.3. We have 1◁H′_{9,0}◁H_{9,0}◁H_{9}◁G_{9}, where G_{9}/H_{9} ≃ S_{9}, H_{9}/H_{9,0} ≃ ℤ, H_{9,0}/H′_{9,0} ≃ (ℤ ⊕ ℤ/3ℤ)^{8}, H′_{9,0} = H′_{9} ≅ ℤ/2ℤ. Proof. [MoTe10], Proposition 2.4.

7. A more basic invariant: braid monodromy factorizations related to branch curves

The first step in computing fundamental groups of complements of a curve is computing their braid monodromy. In fact in order to really understand the group,

14 M. TEICHER

one has to compute braid monodromy factorizations of Δ^{2} (the central element of braid groups related to the curve).

There are many interesting questions which are still open. They include:

Is the data in a braid monodromy factorization of Δ^{2}, related to a branch curve, enough in order to distinguish between different connected components of moduli space? In recent research of V. Kulikov and the author ([KuTe]), it was proved that if 2 curves have equivalent braid monodromy factorizations, the pairs (P_{i}^{2}, B_{i}) are homomorphic and the surfaces (in case of branch curves) are diffeomorphic.

Will equivalent braid monodromy indicate deformation type within the algebraic surfaces category or outside of it? Is π_{2} over π_{1} needed for this purpose?

How does one distinguish between two type braid monodromy factorizations: those induced from algebraic curves and those induced from other curves (see [Mo2])?

How does one determine if 2 braid monodromy factorizations are equivalent? Can one use braid monodromy factorizations to find simplectic invariants (see

[CKTe])?

References on diffeomorphism types can be found in [Ma] and [C].

References

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[CaTo] Carlson J., Toledo D., Discriminant complements and kernels of monodromy representations, to appear in Duke Math. J., alg-geom/9708002, 22pp.

[C] Catanese F., Some old and new results on algebraic surfaces, Proceedings of the First European Congress of Mathematicians, Progress in Mathematics, vol. 119, Birkhäuser, 1994.

[CKTe] Catanese F., Katzarkov L., Teicher M., Symplectic invariants, in preparation.

[CiMT] Ciliberto C., Miranda R., Teicher M., Braid monodromy branch curves of K3-surfaces, in preparation.

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[FRoTe] Freitag P., Robb A., Teicher M., The fundamental group of Galois covers of Hirzebruch surfaces, in preparation.

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[Ma] Manetti M., On some components of moduli spaces of surfaces of general type, Compositio Mathematics 92 (1994), 285-297.

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[MoRoTe] Moishezon B., Robb A., Teicher M., On Galois covers of Hirzebruch surfaces, Math. Annalen 305 (1996), 493-539.

[MoTe1] Moishezon B., Teicher M., Existence of simply connected algebraic surfaces of positive and zero indices, Proceedings of the National Academy of Sciences, United States of America 83 (1986), 6665-6666.

[MoTe2] Moishezon B., Teicher M., Simply connected algebraic surfaces of positive index, Invent. Math. 89 (1987), 601-643.

[MoTe3] Moishezon B., Teicher M., Galois coverings in the theory of algebraic surfaces, Proc. of Symp. in Pure Math. 46 (1987), 47-65.

[MoTe4] Moishezon B., Teicher M., Braid group technique in complex geometry I, Line arrangements in CP^{2}, Contemp. Math. 78 (1988), 425-555.

[MoTe5] Moishezon B., Teicher M., Finite fundamnetal groups, free over Z/cZ, for Galois covers of CP^{2}, Math. Ann. 293 (1992), 749-766.

[MoTe6] Moishezon B., Teicher M., Braid group technique in complex geometry II, From arrangements of lines and conics to cuspidal curves, Algebraic Geometry, Lecture Notes in Math., vol. 1479, 1990, pp. 131-180.

[MoTe7] Moishezon B., Teicher M., Braid group techniques in complex geometry III, Projective degeneration of V_{3},, Contemporary Math. 162 (313-332).

[MoTe8] Moishezon B., Teicher M., Braid group techniques in complex geometry IV, Braid monodromy of the branch curve S_{3} of V_{3} → CP^{2} and application to π_{1}(CP^{2} − S_{3}, ∗), Contemporary Math. 162, 332-358.

[MoTe9] Moishezon B., Teicher M., Braid group techniques 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 (1996), no. 11, 1-120.

[MoTe10] Moishezon B., Teicher M., Fundamental groups of complements of curves in CP^{2} as solvable groups, Israel Math. Conf. Proc. 9 (1995), 329-346.

[Ro] Robb A., The Topology of Branch Curves of Complete Intersections, Ph.D. dissertation, Columbia University, 1994.

[RoTe] Robb A., Teicher M., Applications of braid group techniques to the decomposition of moduli spaces, new examples, Topology and its Applications 78 (1997), 143-151.

[SW] Seiberg N., Witten E., Electric-magnetic duality, monopole condensation, and confinement in N = 2 supersymmetric Yang-Mills theory, Nuclear Physics B 426 (1994), 19-52.

[Si] Simpson C., A relative notion of algebraic Lie group and applications to n-stacks, Algebraic Geometry, 1996.

[Te1] Teicher M., On the quotient of the braid group by commutators of transversal halftwists and its group actions, Topology and its Applications 78 (1997), 153-186.

[Te2] Teicher M., Projective complements of branch curves of Veronese generic projection, in preparation.

[Te3] Teicher M., Chern classes of fibered products, preprint.

[To] Toledo D., Projective varieties with non residual finite fundamental group, Extrait de Publications Mathematiques 771 (1993).

[VK] Van Kampen E.R., On the fundamental group of an algebraic curve, Ann. J. Math. 55 (1933), 255-260.

[W] Witten E., Monopoles and four-manifolds, Math. Research Letters 1 (1994), 769-796.

[Z] Zariski O., Algebraic Surfaces, (Ch. VIII), Second Edition, Springer, 1971.

Department of Mathematics, Bar-Ilan University, 52900 Ramat-Gan, Israel

   E-mail address: teicher@macs.biu.ac.il