Braid Monodromy of Algebraic Curves

These are the notes from a one-week course on Braid Monodromy of Algebraic Curves given at the Université de Pau et des Pays de l’Adour during the Première Ecole Franco-Espagnole: Groupes de tresses et topologie en petite dimension in October 2009. This is intended to be an introductory survey through which we hope we can briefly outline the power of the concept monodromy as a common area for group theory, algebraic geometry, and topology of projective curves. The main classical results are stated in §2, where the Zariski–van Kampen method to compute a presentation for the fundamental group of the complement to projective plane curves is presented. In §1 these results are prefaced with a review of basic concepts like fundamental groups, locally trivial fibrations, branched and unbranched coverings and a first peek at monodromy. Descriptions of the main motivations that have lead mathematicians to study these objects are included throughout this first chapter. Finally, additional tools and further results that are direct applications of braid monodromy will be considered in §3. While not all proofs are included, we do provide either originals or simplified versions of those that are relevant in the sense that they exhibit the techniques that are most used in this context and lead to a better understanding of the main concepts discussed in this survey. Nothing here is hence original, other than an attempt to bring together different results and points of view. It goes without saying that this is not the first, and hopefully not the last, survey on the topic. For other approaches to braid monodromy we refer to the following beautifully-written papers [73, 20, 6]. We finally wish to thank the organizers and the referee for their patience and understanding in the process of writing and correcting these notes. Acknowledgment. The author was partially supported by the Spanish Ministry of Education MTM2007-67908-C02-01 and MTM2010-21740C02-02.


Settings and Motivations
For the sake of completeness we will define the main objects and will state the problems that motivate the study of braid monodromy in connection with algebraic curves.
Remark 1.2. Also note that if x 0 and y 0 are in the same path-connected component of X, then the groups π 1 (X, x 0 ) and π 1 (X, y 0 ) are naturally isomorphic by an inner automorphism. In case X is path connected, such groups are denoted by π 1 (X) and called the fundamental group of X. Example 1.3. π 1 (S 1 ) = Z (see the comment after Theorem 1.14).
Note that Y n ϕ ∼ = X n /Σ n , where Σ n represents the action of the symmetric group of n elements on X n by permuting the coordinates, that is, if σ ∈ Σ n , then σ(z 1 , ..., z n ) = (z σ (1) , ..., z σ(n) ) (note that the elements of X n /Σ n are simply sets of n distinct complex numbers). The homeomorphism ϕ is given as follows: any polynomial f (z) ∈ Y n can be normalized as f (z) = (z −z 1 ) · · · (z −z n ) where z i = z j . Thus ϕ(f ) := {z 1 , ..., z n } ∈ X n /Σ n . Conversely, given a set of n distinct complex numbers {z 1 , ..., z n } ∈ X n /Σ n one can obtain f (z) = z n + a n−1 z n−1 + · · · + a 1 z + a 0 as a i = σ n−i (z 1 , ..., z n ) = the symmetric polynomial of degree n − i on z 1 , ..., z n . Therefore, if γ is a path in Y n from f 1 = (z − x 1 ) · · · (z − x n ) to f 2 = (z − y 1 ) · · · (z − y n ), then ϕγ can be seen as a collection of n disjoint paths γ i , i = 1, ..., n from x i to y σ(i) for a certain σ ∈ Σ n . Then π 1 (Y n ) = B n , the (geometric) braid group on n strings (on C ).
In the previous examples fundamental groups are either computed directly or by finding suitable homomorphisms to other spaces whose fundamental group was easier to compute. The idea behind it is that the fundamental group is a topological invariant, that is, if X ϕ ∼ = Y are two homeomorphic spaces, then the map π 1 (X; x 0 , x 1 ) ϕ * →π 1 (Y ; ϕ(x 0 ), ϕ(x 1 )) given by the set-theoretical image by ϕ of paths in X is well defined, it is a bĳection for any choice of x 0 , x 1 ∈ X, and it preserves the products, hence it is an isomorphism in the category of groupoids. In particular, ϕ * defines isomorphisms of fundamental groups.
However, homeomorphisms are not the only continuous maps that induce isomorphisms of fundamental groups. The following result generalizes the map ϕ * referred to in the previous paragraph and it serves as a way to introduce notation. Its proof is straightforward from the definitions and it is left as a useful exercise for the beginners. Lemma 1.6. Any continuous map ϕ : X → Y between two topological spaces induces morphisms ϕ x 0 ,x 1 : π 1 (X; x 0 , x 1 ) → π 1 (Y ; ϕ(x 0 ), ϕ(x 1 )) for any choice of x 0 , x 1 ∈ X.
To simplify notation, and whenever there is no likely ambiguity, we will simply refer to ϕ x 0 ,x 1 as ϕ * .
Example 1.7. Assume that Y ⊂ X and that there is a surjective continuous map ϕ : X → Y such that Y i →X ϕ →Y is the identity on Y . Then ϕ * is an epimorphism, since ϕ * • i * = (Id Y ) * (see 1.6) which is an isomorphism. Such a map is called a retraction of X onto Y .
The following is a very common way to find maps that induce equivalent morphisms of fundamental groups. Definition 1.8. Let f, g : X → Y two continuous maps. We say that f and g are two homotopic maps if there exists H : X × [0, 1] → Y a continuous map such that, if x ∈ X then H(x, 0) = f (x) and H(x, 1) = g (x). The map H is called a homotopy from f to g and it is denoted as f H ∼ g. Two topological spaces X and Y are called homotopy equivalent if there exist maps f : X → Y and g : Y → X such that f •g ∼ Id Y and g•f ∼ Id X Example 1.9. If two topological spaces X and Y are homotopy equivalent, then their groupoid fundamental groups are isomorphic.
Note, in particular, that a homotopy equivalence ϕ : X → Y induces isomorphisms of fundamental groups π 1 (X; x) ϕ * →π 1 (Y ; ϕ(x)). Moreover, if both spaces are connected, then one can simply say that π 1 (X) 10. Assume the hypothesis of Example 1.7 and also assume that the retraction ϕ is homotopic to the identity in X. Then X and Y are homotopy equivalent and the retraction ϕ is an equivalence of homotopies. Such retractions are called deformation retract. Example 1.11. π 1 (C \{0}) = Z is a consequence of Examples 1.3 and 1.10, since the normalization map C \{0} → S 1 given by z → z |z| is a deformation retract.

The Seifert-van Kampen Theorem
One of the basic tools to compute fundamental groups (and fundamental groupoids) is the Seifert-van Kampen Theorem. This was first proved by H. Seifert [70] and later on, and independently, by E.R. van Kampen [41]. Originally van Kampen wrote this paper in an attempt to prove that the construction O. Zariski [76] built in order to compute the fundamental group (aka Poincaré group) of the complement of a plane curve in P 2 was correct.
In order to state this result we will need to define the amalgamated product of two groups. Definition 1.12. Let G 12 i 1 →G 1 and G 12 i 2 →G 2 be two group homomorphisms. The amalgamated free product of G 1 and G 2 w.r.t. G 12 is a group G that fits in a commutative diagram and has the following universal property: for any other such G there exists a homomorphism G ϕ →G that commutes with both diagrams, that is, ϕj 1G = j 1G and ϕj 2G = j 2G . This can also be described by saying that the diagram (1.1) is a pushout (in the category of groups).
In more down-to-earth terms, if G 12 , G 1 , and G 2 are groups is as in the previous definition with morphisms i 1 , i 2 respectively, then the amalgamated free product of G 1 and G 2 w.r.t. G 12 , commonly denoted by G 1 * G 12 G 2 , can be described as the quotient Therefore, if G i , are finitely presented, and G 12 is finitely generated, then G 1 * G 12 G 2 is finitely presented.

147
We will give the following version of the main theorem. Theorem 1.14 (Seifert-van Kampen Theorem). Let U 1 and U 2 pathconnected open subsets of X such that: Then . In other words, the commutative diagram given by the inclusions: Originally van Kampen considered the general case scenario, where the open sets U 1 , U 2 and U 12 are not necessarily path-connected. In this case, the result above generalizes claiming that π 1 (X, x 0 , y 0 ) is a pushout of π 1 (U 1 , x 0 , y 0 ) and π 1 (U 2 , x 0 , y 0 ) in the category of groupoids (see [13, 6.7.2]).
This theorem gives a very simple proof of Example 1.3 (see [13, 6.7.5] By induction, if n S 1 := S 1 ∨ ... ∨ S 1 is the bouquet of n spheres, then π 1 ( n S 1 ) = F n . Example 1. 16. Let z 1 , ..., z n ∈ C , Z n := {z 1 , ..., z n }. Then π 1 (C \Z n ) = F n . The case n = 1 is shown in Example 1.11. The case n = 2 is given in Figure 1.1 by describing S 1 ∨ S 1 as a deformation retract of C \ {±1}. In general, one can describe n S 1 as a deformation retract of C \ Z n , and hence the result will follow from Examples 1.10 and 1.15.

Locally Trivial Fibrations
Definition 1.18. A surjective smooth map π : X → M of smooth manifolds is a locally trivial fibration if there is an open cover U of M and diffeomorphisms ϕ U : π −1 (U ) → U × π −1 (p U ), with p U ∈ U , such that ϕ U is fiber-preserving, that is pr 1 • ϕ U = π. The diffeomorphisms ϕ U are called trivializations of π. The submanifold π −1 (p) ⊂ X is called the fiber of π at p and usually denoted by F p .
Two fibrations π : X → M , π : X → M are said to be equivalent if there exists a diffeomorphism ϕ : X → X such that . Therefore, the existence of the points p U ∈ U in Definition 1.18, might be replaced by the same property at any point of U . Hence all the fibers of a locally trivial fibration are all diffeomorphic to F p as long as X is connected. Example 1.20. Any product X := M × F produces a locally trivial fibration just by projecting onto a component, say π : The open cover of M that trivializes the fibration is given simply by the total space M . The fiber of this fibration at any point is isomorphic to F . Such a fibration is called a trivial fibration.

Unbranched Coverings, Branched Coverings, and Monodromy
We will briefly discuss the notion of unbranched and branched coverings as both, a motivation and a first approximation to braid monodromy. Conditions for the existence of branched coverings of smooth lines and surfaces ramified along a given locus has been a classical problem that becomes a common place for (low dimensional) Topology, Algebraic Geometry, (inverse) Galois Theory, and Geometry. The main results of this section can be found in much more detailed in [59]. An unbranched covering is a locally trivial fibration whose fiber is a discrete subset.
Example 1.26. The map π : B * → B * , defined as π(z) = z e from the punctured disc to itself is a finite unbranched covering whose fiber is a finite set of e elements. In particular, . Analogously, the map π : B n−1 × B * → B n−1 × B * defined as π(z 1 , ..., z n−1 , z n ) = (z 1 , ..., z n−1 , z e n ) is also an unbranched covering whose fiber is a finite set of e elements.
(for a proof of Theorem 1.27 see any basic textbook on Algebraic Topology, for instance see [75]). Example 1.28. Note that Example 1.26 induces the following: where the map is given by π * (γ) = eγ. This corresponds to the inclusion Ze < Z. Analogously, the map π : B n−1 × B * → B n−1 × B * defined as π(z 1 , . . . , z n ) = (z 1 , . . . , z n−1 , z e n ) induces the following: where the map is also given by π * (γ) = eγ. This corresponds to the inclusion Ze < Z as shown below.

.2. Monodromy of Unbranched Coverings
Any unbranched covering π : X → M is, by definition, a locally trivial fibration whose fiber is a discrete set S := π −1 (q 0 ). There is a monodromy action of π 1 (M, q 0 ) on S as follows. Let γ : [0, 1] → M be a closed path in π 1 (M, q 0 ). One has the following diagram According to Example 1.23,π is a trivial fibration. The trivialization ofπ defines a bĳection γ : S → S. In other words, for any given x 0 ∈ S one can construct a section s x 0 : [0, 1] →X such that s x 0 (0) = x 0 (see is the section constructed above and hence is the image of z by the monodromy of γ. Analogously, note that s ξ i z (λ) = defines the monodromy of γ on S, which is just a cyclic transformation of order e.

Branched Coverings
In this section we will focus on the study of branched coverings of complex manifolds.

Definition 1.30.
Let M be an m-dimensional (connected) complex manifold. A branched covering of M is an m-dimensional irreducible normal complex space X together with a surjective holomorphic map π : X → M such that: • every fiber of π is discrete in X, • the set R π := {x ∈ X | π * : O π(x),M → O x,X is no isomorphism} called the ramification locus, and B π = π(R π ) called the branched locus, are hypersurfaces of X and M , respectively, • the map π| : X \ π −1 (B π ) → M \ B π is an unramified (topological) covering, and • for any q ∈ M there is a connected open neighborhood W q ⊂ M such that for every connected component U of π −1 (W ): (1) π −1 (q) ∩ U has only one element, and (2) π| U : U → W is surjective and proper.
Example 1.31. The map π : B → B defined by π(z) = z e is a branched covering ramified at B π = {0}. Analogously, the map π : B n → B n defined by π(z 1 , . . . , z n ) = (z 1 , . . . , z n−1 , z e n ) is a branched covering ramified at B π = {z n = 0}. The purpose of this section is to study Theorem 1.27 (2) for branched coverings, that is, what are the conditions, in terms of the monodromy or in terms of fundamental groups for the existence of branched coverings ramified along a given divisor. In order to do so, let us develop the key concept of meridian.
for a certain b ∈ B as above (see Figure 1.2).

Proposition 1.34. Any two meridians, say
Moreover, the conjugacy class of a meridian coincides with the set of homotopy classes of meridians around the same irreducible component.
Proof. The main ingredient of this proof is that B \ Sing(B) is a path connected space as long as B is irreducible since Sing(B) has real codimension 2 in B. Therefore consider δ a path in B fromb 2 tob 1 , where One can deform δ along the normal bundle so that δ connectsb 2 andb 1 . This way, note that The moreover part is obvious by definition of meridian. If γ = α · γ b · α −1 is a meridian decomposed as in Definition 1.33, and ω ∈ π 1 (M \ B, q 0 ) then (ω · α) · γ b · (ω · α) −1 also satisfies the conditions of Definition 1.33, and hence it is a meridian of M around B .

Existence and construction of branched coverings: smooth case
Consider B a non-singular hypersurface, . According to Proposition 1.34, Jē does not depend on the choice of the meridians. Definition 1.35. Under the above notation, π is said to ramify (resp. ramify at most) along D if B is the ramification locus of π and e i coincides with (resp. is a multiple of) the ramification index of π at D i .
A branched cover π ramified along D is said to be maximal if it factors through any other branched cover π ramified at most along D, that is, there exists a holomorphic map ϕ : X → X such that: Remark 1.36. Note that if π : X → M is a branched covering ramified along D, then γ e i i can be lifted to a meridian of π −1 (D i ) (see Remark 1.32 and Example 1.28). Therefore Jē π 1 (X \ π −1 (B), q 0 ).
The following result can be found in [59,Theorem 1.2.7]. It characterizes the branched covers of a complex manifold ramified along a smooth hypersurface with prescribed ramification indices and it is a partial equivalent of Theorem 1.27 (2).
There is a natural one-to-one correspondence between As a consequence of Theorem 1.37 one has the following classical result, for compact complex manifolds of dimension 1, part of which is known as the Riemann Existence Theorem. Consider M g a compact complex manifold of dimension 1, that is, a Riemann surface and Z n ⊂ M g a finite set of n points in M g . Theorem 1.38. Any monodromy action π 1 (M g \Z n ) → Σ s can be realized by a branched covering of the Riemann surface M g .
For any meridian γ z of an element of z ∈ Z n , consider µ(γ z ) ∈ Σ s . Since Σ s is finite, the order of µ(γ z ), say e z , is also finite. Define B = {z ∈ Z n | e z > 1} and D = z∈B e z z.
Note that K f.i π 1 (M \ B) and K ⊃ Jē by construction. All one needs to check is condition (1.4.6), but this is also immediate. If γ d z ∈ K, then µ(γ z ) d = 1. Therefore e z |d, since e z is the order of µ(γ z ). Finally, one can apply Theorem 1.37, since B is a smooth hypersurface.

Existence and construction of branched coverings: general case
We will follow the notation introduced in the previous item. Consider Let γ 1 , . . . , γ r ∈ π 1 (M \ B) be meridians of the irreducible components of B and define Jē as above.
In addition, for any q ∈ Sing B one can consider the inclusion of a local neighborhood of q in B, say i q : W q \ B → M \ B. By the special structure of analytic singularities (see [55,Theorem 2.10]), it turns out that i q does not depend on W q for a small enough neighborhood. Therefore, given any It is reasonable, but not so obvious anymore, that given a branched cover π : X → M ramified along D, then K = π * (π 1 (X \ π −1 (B))) satisfies 1.4.8 (see [59,Theorem 1.3.8] or [37, p.340] for a proof).

Theorem 1.39. There is a one-to-one correspondence between
and satisfies (1.4.6) and (1.4.8).
This will allow for a general study of branched covers of P 2 ramified along plane curves, which is the classical problem, already stated by Enriques [29], Zariski [76,77], and many others, known as the multiple plane problem. The original problem was stated as follows: In order to do so, one needs to compute the fundamental group π 1 (P 2 \ (D 1 ∪D 2 )). This will be presented in a more systematic way in Chapter §2. You can go ahead read it and come back, or just bare with me a couple of calculations and hopefully everything will be understood later.
The space Avoiding tangencies at infinity will make our life easier in this case, so one can change the affine coordinate system and simply work with the curve C := {27y 2 = 4(x − y) 3 }. Since this transformation is continuous. The fundamental group is not affected by that. First of all note that C has only one singular point at (0, 0). Consider the projection (x, y) → x, and note that, when restricted to C, it produces a cover of C branched along x = 0 (the projection of the singular point) and x = 1 (the tangency shown by the blue line). Precisely the non existence of vertical asymptotes will allow us to take big disks D Consider γ 1 , γ 2 ,γ 2 meridians around p 1 , p 2 , andp 2 respectively. One can check that these meridians satisfy the following relations as closed paths in the total space C 2 \ D 1 :γ Moreover, where γ ∞ is a meridian of D 2 , the line at infinity of P 2 .
According to Theorem 1.39 we need to study subgroups Jē normally generated by γ e 1 1 , γ e 1 2 , and γ e 2 ∞ forē = (e 1 , e 2 ) ∈ N 2 . Equivalently, one can study quotients of π 1 (P 2 \ (D 1 ∪ D 2 )) of the form [23]) and Gē is finite if and only ifē = (2, 2), (3,4), (4,8), (5,20) or (6,2). In which cases one has the following result (c.f. [ However, there is no maximal Galois cover of P 2 ramified along D = 6D 1 + 2D 2 . Analogously to the Riemann Existence Theorem 1.38, one has the following result on the existence of branched covers ramified along divisors with prescribed ramification index. The proof of this result is similar to the one presented here for Theorem 1.38 and it relies on the fact that π 1 (M \B) is finitely generated, which is a consequence of the Zariski Theorems of Lefschetz Type (see §1.7) and §2.

Chisini Problem
In this context, another interesting motivation is the following problem: Problem 1.4.11. Chisini Problem [19] Let S be a non-singular compact complex surface, let π : S → P 2 be a finite morphism having simple branching, and let B be the branch curve; then "to what extent does the pair (P 2 , B) determine π"?
Partial results have been given to this problem for generic coverings [57,46,45,60], or special types of singularities [44,54], but a global answer to this is yet to be determined. Certain restrictions, like the fact that the degree of the covering has to be ≥ 5, are also known [56,15].

Monodromy Action on Fundamental Groups
Probably the first appearance in the literature of this fact is due to Chisini [18], and has been implicitly used by van Kampen [42] and Zariski [76] in the context of computing the fundamental group of plane projective curve complements. The first systematic approach for the case of plane curves is given by Moishezon [57] with the purpose of studying the Chisini Conjecture.
In order to give a general definition in our setting let us recall the notion of section. Definition 1.43. Let π : X → M be a locally trivial fibration. We say that a morphism s : Associated with a locally trivial fibration π : X → M and a section s : M → X there is a right action of the groupoid {π 1 (M, p 1 , p 2 )} on the groups {π 1 (F, q 0 )}, called monodromy action of M on F . More specifically, given a path γ ∈ π 1 (M, p 1 , p 2 ) (s(p 1 ) = (p 1 , q 1 ), s(p 2 ) = (p 2 , q 2 )) and a closed path α ∈ π 1 (F, q 1 ), one obtains another closed path α γ ∈ π 1 (F, q 2 ).

Construction of the monodromy
Consider γ an open path representing an element in π 1 (M, p 1 , p 2 ). The following diagram comes from restriction: (1) The mapπ is a fibration which, by Example 1.23, is trivial, and hence consider a trivialization [0, 1] × F ϕ −→X and a sections : (2) Any path α ∈ π 1 (F, q 1 ) can be regarded as a path α : constructed above is called the monodromy action of γ over α. Remark 1.45. Intuitively, α is being pushed fiberwise along γ and keeping the base point along the section s. One needs to check that the previous construction is independent of ϕ, the choice of representative of γ and α. This is all a consequence of the Homotopy Lifting Property.
Note that, according to our discussion, The following example will clarify the previous construction.
Example 1.46. The trivial case occurs when q 0 := q 1 = q 2 , the fibration π is trivial, and the section s : M → M × F is given by s(p) = (p, q 0 ). In this case, the monodromy action is trivial.   (1) In this case, the closed paths α 1 , α 2 shown in the previous example are transformed as shown in Figure 1.8,

Mapping Class Groups and Braid Action
The group of oriented isomorphisms of a compact orientable surface S of genus g fixing a set of n points up to homotopy relative to its boundary is called the mapping class group of S g n , and will be denoted by M(S g n ). A classical interpretation of the geometric braid group on n-strings (see Example 1.5) is the following. Theorem 1.49 ([12]). There is an isomorphism between the geometric group of braids on n-strings and the mapping class group of the disk D fixing a set of n points, that is, This allows one to interpret the action of the braid group on free groups as a monodromy action.  Note that the trivializations are nothing but the isotopy that joins a diffeomorphism and the identity.
Using Proposition 1.51 and Theorem 1.49 one can consider the action, via monodromy, of a braid in B n on π 1 (D \ Z n ) = F n = Zg 1 * ... * Zg n .
It is an interesting exercise to convince oneself that the monodromy action of a standard basis σ 1 , . . . , σ n−1 on g 1 . . . , g n is given as follows: (1.5) This is basically a consequence of Example 1.48 and Figure 1.8.
where D is the disk centered at 0 of radius 2, defined by (x, y) → x. Note that π is a proper submersion, and hence a locally trivial fibration by the Ehresmann Fibration Theorem 1.24.
Therefore, which corresponds to the braid σ := σ 1 σ 2 · · · σ k−1 . Note that (1.6) Example 1.55. Based on Example 1.54 one can generalize this construction to study the monodromy of the fibration π : It is easy to see that such monodromy is nothing but p times the monodromy of π : X = D * × D \ {y q = x} → M = D * , which corresponds to the braid (σ 1 σ 2 · · · σ q−1 ) p . In particular, one can recuperate the result given in Example 1.53 for q = 2, p = k.

Zariski Theorem of Lefschetz Type
From the previous sections one fact seems to be worth stressing: In order to understand coverings of M ramified along D one needs to study π 1 (M \ B).
How to compute the fundamental group π 1 (M \B) of a quasi-projective variety? The following crucial result, known as the Zariski Theorem of Lefschetz Type (cf. [38,33]) states that it is enough to understand complements of curves on surfaces.
For this reason, we will be mostly concerned about complements of projective curves in the complex plane P 2 . However, it is important to stress that the general problem of computing homotopy groups of complements to singular varieties and relating them to other invariants of the complement is a very interesting question in and of its own (see [65,51,52]).

Zariski-van Kampen Method
The Zariski-van Kampen method allows us to give a finite presentation for the fundamental group of the complement to a projective plane curve. Originally sketched by Zariski [76] and later completed by van Kampen [42]. Later on, D. Chéniot [16] gave a modern approach to this method. The method is constructive and in some cases it is even effective, i.e. it has been implemented in the case of line arrangements, curves with easy singularities and equations on the Gaussian integers Z[ √ −1] (see [14,11]). A very nice approach to this method can be found in the unpublished notes written by I.Shimada in [73].
We will put together several ingredients, among which, the van Kampen Theorem is key.

Fundamental Group of the Total Space of a Locally Trivial Fibration
Let π : X → M be a locally trivial fibration with section s : M → X.
Consider p ∈ M and x 0 ∈ F p . (M, p), where the action of π 1 (M, p) on π 1 (F p , x 0 ) is given by the monodromy of π.
Besides Proposition 1.34, we need another basic result on meridians.

Zariski-van Kampen Theorem
Let C ⊂ P 2 be a projective plane curve given as the zeroes of a reduced homogeneous polynomial f ∈ C [X, Y, Z] of degree d. After a suitable change of coordinates one can assume P = [0 : 1 : 0] ∈ P 2 \ C and thus one can consider the projection π : P 2 \ {P } → P 1 from P . Note that, for any point z = [x 0 : z 0 ] the preimage π| C consists of a finite number of points, precisely the roots of the one-variable polynomial f (x 0 , t, z 0 ).
This implies that π| C is a branched cover of P 1 of degree d ramified In other words, π| C ramifies along those points of P 1 whose vertical lines above them intersect C in less than d distinct points (see Figure 2.2).
The projection from P Let L := L 1 ∪· · ·∪L n be the union of the non-generic vertical lines, that is, L := π −1 (∆). Even though π is a locally trivial fibration, there are two problems: first of all it is not so obvious since all the fibers are very close to P and second of all, the fiber is NOT π −1 ([x 0 : z 0 ]) ∼ = P 1 \{P }. We would like to separate the fibers. In order to do so one can construct another complex space X from P 2 by replacing P by the P 1 of lines passing through P . In other words, each line L q := π −1 ([x 0 : z 0 ]) will be compactified not by adding P , but by adding a point P q . Algebraically this can be done as follows. Consider U Y = {[X : Y : Z] | Y = 0} an affine chart of P 2 containing P and define the following map ε : U Z using the same transition functions as for U Y , U X , and U Z . This way one defines the manifold X. Nowπ = π • ε can be extended to X as follows According to (2.1) one can check thatπ| X\E = π| P 2 \{P } . Moreover, if Hence L q ∪ {P q := ([0 : 1 : 0], [z 0 : x 0 ])} andL q ∼ = P 1 . DefineC := ε −1 (C),L := ε −1 (L) the preimages of C and L, respectively, by the blow-up. Note thatC ∼ = C by the above discussion, since P / ∈ C. Also, note thatπ|C is a branched cover andπ| (C∪L) is an unbranched cover.
Proposition 2.5. The group π 1 (P 2 \ (C ∪ L)) is generated by g 1 , . . . , g d , γ 1 , . . . , γ n and a finite set of defining relations is given by: γ n · · · γ 1 = 1, g d · · · g 1 = 1 and g It is an immediate consequence of Theorem 2.1 and Proposition 2.4. For the sake of simplicity, note that we have replaced s * (γ j ) simply by γ j . The relations coming from the monodromy should read Finally, one can give a presentation of π 1 (P 2 \ C) as follows.
As an immediate application of this Theorem one has the following. Proof. We will use the fact that H 1 (X) = π 1 (X)/[π 1 (X), π 1 (X)], that is, the first homology group H 1 (X) of a topological space X is the abelianization of π 1 (X) its fundamental group (see for instance [58,Lemma 11.69.3]).
First of all, by Proposition 1.34, we know that H 1 (P 2 \ C) is a quotient of Z r , since π 1 (P 2 \ C) is generated by meridians of the irreducible components of C and any two meridians of the same irreducible component are conjugated.
Finally, and this is the key here, Theorem 2.6 specifies that the quotient of Z r mentioned above comes from abelianizing the relations g d · · · g 1 = 1, g γ j i = g i , j = 1, . . . , n − 1. By construction of the monodromy, the element g γ j i is a meridian around the same irreducible component as g i , hence these relations are trivial in H 1 . The only relation left is g d · · · g 1 = 1. Note that in the set {g 1 , . . . , g d } there are exactly d i meridians of the component C i , hence, after abelianizing, g d · · · g 1 = 1 becomes where m 1 , . . . , m r represent cycles around the component C 1 , . . . , C r respectively. Therefore which has rank r−1 and non-trivial torsion τ if and only if τ = (d 1 , ..., d r ) > 1.
Remark 2.9. The projection π used for the Zariski-van Kampen method as presented here is only required to be performed from a point P / ∈ C. Originally, π was asked to be generic in the following sense: (1) Any line L through P contains at most one singular point of C or one tangency, (2) no lines through P are higher order tangents at a smooth point of C, and (3) any line L through P that intersects C at a singular point Q satisfies that mult Q (C) = mult Q (L, C).
Geometrically, this means that the following cases are avoided in the locally trivial fibration π: Obviously, one can always chose P so that π is generic, since the set of higher order tangencies at C, lines containing more that one singular point, bitangencies, and lines in the tangent cone of a singularity of C is finite, so P can be chosen outside this set and C.
The following, very natural, result assures that if two curves can be joint by a smooth path of equisingular curves, then their fundamental groups are isomorphic (see [14] for a proof).

Proposition 2.10. All curves in the same connected family of equisingular curves are isotopic.
One also has results on how the fundamental group of a family of equisingular curves changes when degenerating onto other curves outside the equisingular locus. ([25]). Let {C t } t∈(0,1] be a continuous family of equisingular curves degenerating onto the reduced curve C 0 . Then there is a natural epimorphism

Proposition 2.11
In particular, Corollary 2.12. If C can be continuously degenerated onto a curve with abelian fundamental group, then π 1 (P 2 \ C) is abelian as well.
Other interesting degeneration results can be found in [5].

The fundamental group of smooth and nodal curves
First, we will compute the fundamental group of the curve using the Zariski-van Kampen method described above.
(2) The projection π : P 2 \C → P 1 ramifies along ∆ : After blowing up, the projectionπ|π−1 (P 1 \∆) is a locally trivial fibration of fiber P 1 \ Z d . Note that this projection is highly not generic, since each non-generic fiber, , and by induction g 1 = ... = g d = g. Finally, g 1 · · · g d = 1 becomes g d = 1. Therefore, Note that all necessary relations are obtained by the monodromy action of any meridian γ i . This can be further improved.

Theorem 2.13. If C is an irreducible curve with a maximal order tangent, that is, if there exists a line L such that
Proof. Consider P ∈ L, P = Q and project from P . Since L becomes a non-generic fiber of the projection, one can fix a base point x 0 on P 1 sufficiently close to the projection of L, say z 1 . The monodromy around z 1 is given as in Example 1.54. The computation above shows that the relations obtained from this monodromy are enough to verify that π 1 (P 2 \ C) is abelian.
Another application of the computations above. Theorem 2.14. If C is a smooth curve of degree d, then π 1 (P 2 \C) = Z/dZ.
Proof. The family of smooth curves of degree d is a quasi-projective variety in the projective space P N of dimension N = d− 2 2 , where N + 1 is the number of coefficients of a generic homogeneous polynomial of degree d in C [X, Y, Z]. Therefore it is path connected, and, by Proposition 2.10, it is enough to compute the fundamental group of a particular smooth curve of degree d. The curve C defined above is smooth since

Hence (2.3) gives the required fundamental group.
The simplest singularities a curve can have are nodes (aka. ordinary double points), that is, singular points that admit local equations of the form x 2 + y 2 , where x and y are generators of the local ring O P 2 ,P . Note that x 2 +y 2 is equivalent to x 2 −y 2 = (x−y)(x+y) by a complex change of coordinates. In other words, a node locally looks like a product of smooth transversal branches (locally meaning inside a neighborhood of the point, as shown below).
A more general result regarding nodal curves was already given by Zariski [76,Theorem 7]. Theorem 2.15 (Zariski, Fulton, Deligne, Salvetti). Any nodal curve has an abelian fundamental group. Remark 2. 16. As in our proof of Theorem 2.14 Zariski's proof of Theorem 2.15 depended on the irreducibility of the moduli spaces of nodal curves (there are different strata depending on the number and degrees of irreducible components). Such result had been claimed by Severi [72, Anhang F], and hence the proof given by Zariski was completed. However, later on, a gap was found in Severi's proof and hence Zariski's result was not complete anymore. Severi's assertion became Severi's problem and the original result by Zariski turned into the Zariski conjecture on nodal curves and they remained open until 1980, when Fulton [32] first and then Deligne [24] proved the Zariski conjecture on nodal curves (giving algebraic and topological proofs respectively) without using Severi's result. Finally, in 1986, J. Harris [39] solve the Severi problem. For a further study of such problems see [63,74,34,35,36] among others.
One can also find more recent proofs of this result by means of monodromy computations (see M.Salvetti [68]).
Also, generalizations of this result have been proved by M.V.Nori in [61].
The same ideas in Deligne's proof lead to the following result.
Proposition 2.17. If C 1 and C 2 are two curves intersecting transversally (only in ordinary double points), then where L is a line tranversal C 1 ∪ C 2 .

Further examples
By the previous sections we know how to compute fundamental groups of all curves of degrees one, two, and three: (1) Degree one: π 1 (P 2 \ L) = {1} (since L is smooth and of degree one).
(3) Degree three: where L i , i = 1, 2, 3, are lines in general position (L 1 L 2 L 3 is a nodal curve union of three smooth curves of degree one).
Proof. Projecting from a point outside the lines one realizes that there is only one special fiber. Since Theorem 2.6 involves the monodromy action of all meridians but one, then there are no relations coming from monodromy, that is, π 1 (P 2 \ (L 1 ∪ L 2 ∪ L 3 )) = g 1 , g 2 , g 3 : g 3 g 2 g 1 = 1 = Z * Z.
(c) π 1 (P 2 \ (Q ∪ L)) = Z, where Q is a conic and L is a line transversal to Q. (d) π 1 (P 2 \ (Q ∪ L)) = Z, where Q is a conic and L is a tangent line to Q.
Proof. Projecting from a point P on L one realizes that there are two special fibers: L and L both tangent lines to Q through P . Consider γ a meridian around the projection of L.
Proof. Since C 3 has an inflection point, one simply applies Theorem 2.13.
Probably the easiest example of non-abelian fundamental group of an irreducible quartic (i.e. a curve of degree four) is the three-cuspidal quartic. Zariski [78] showed this in a more general setting using a brilliant argument. Let us sketch the proof. Theorem 2.18. Let C be a (rational) curve of degree 2d, with 2(d−1)(d− 2) nodes and 3(d−1) cusps. Then π 1 (P 2 \C) = B d+1 (P 1 ) (see Example 1.5).
Proof. Such curves are generic plane sections of the space∆ d+1 of homogeneous polynomials of degree d + 1 in two variables with multiple roots, described in Example 1.5. The reason is the following: a plane in the spacē Y d+1 of homogeneous polynomials of degree d + 1 in two variables is nothing but a family of polynomials E : ⊂ P 2 at a tangent. Note that F has degree d + 1. In fact it is a (rational) curve with d(d−1) 2 nodes. Therefore E ∩∆ d+1 is exactly the dual of F , sayF , which has to have degree 2d, 2(d − 1)(d − 2) nodes and 3(d − 1) cusps.

Braid monodromy of curves: local versus global
When computing the monodromy action of the locally trivial fibration π : X → P 1 \ ∆ constructed in the Zariski-van Kampen method one needs a collection of meridians around the points in ∆. We recall that a meridian γ around z ∈ ∆ can be decomposed as γ = ω · γ z · ω −1 , where ω is a path joining the base point z 0 and a point z near z, and γ z is the boundary of a disk centered at z (see Definition 1.33).
The action of γ on π 1 (F, s * (x 0 )) will also be decomposed as the action of γ z on π 1 (F, s * (z )) and the action of ω on π 1 (F, s * (x 0 ), s * (z )). The first one will be called the local monodromy at z and the second one will be called the global monodromy at z.
The local monodromy is completely determined by the local topological type of the curve on the points on the fiber (see Examples 1.53, 1.55, and 1.54). For instance, the Puiseux expansion at each singular point on the fiber determines the local monodromy.
However, the global monodromy depends on the position of singularities and, in general, it depends on the global geometry of the curve. Whether or not there is a finite set of global data on the curve that determines the global monodromy is still unknown.
In the previous sections only examples were presented where the local monodromy information was enough to give the monodromy action, but this is far from being the case in general. The following example will hopefully depict the general situation.
Example 2.20. Consider the following quartic, which is a union of two smooth conics intersecting at one point: When projecting from [0 : 1 : 0] there are five special fibers After choosing a base point we can start computing the braid monodromy as follows: The tangency and the high order tacnode can basically be obtained directly from the local monodromy, since the global monodromy is trivial (the base point is close enough to both special fibers).
The tangency can be computed directly from Example 1.54 as σ 2 and the tacnode, whose local equation is y 2 = x 8 , that is two smooth branches with multiplicity of intersection 4, can be obtained from Example 1.53 as σ 8 1 . However, the remaining braids depend on global monodromy for two different reasons: • the left-most tangency depends on global monodromy basically due to the fact that the branches from the small conic become complex conjugated and intertwine with the branches of the big conic as one approaches the tangency obtaining the following braid.

185
• the tangent immediately to the right of the tacnode also depends on global monodromy, even though all the branches remain real. The reason in this case is that the approaching path ω consists of half a turn around the tacnode. The braid becomes σ 4 1 · σ 2 · σ −4 1 .
• the right-most tangent also depends on global monodromy for both reasons, one has to avoid the branching values by performing half turns and also the real branches become complex conjugated at some point. However, since one only needs to compute all the monodromy actions but one, this one can be disregarded.
Finally, using (r 2 ), (r 3 ), and (r 8 ) one can easily check that (r 1 ), (r 5 ), (r 6 ), and (r 7 ) become trivial. Therefore, according to the Zariski-van Kampen Theorem 2.6 which is the biggest group whose abelianized is Z⊕Z 2 . This result can also be obtained from the fact that both conics generate a very special pencil with a reduced member, the tangent line to both conics at the tacnode, but that would be another story and it is left to the interested reader.

Definitions and First Properties
LetC = C 0 ∪C 1 ∪...∪C r , be the decomposition in irreducible components of a projective plane curveC. Let us denote by d i the degree of C i and assume C 0 is a transversal line. An alternative construction similar to the Zariskivan Kampen method occurs when studying C 2 := P 2 \C 0 , C :=C ∩C 2 . The space C 2 \C retracts into a compact polydisk minus C as in the  The projection onto the first coordinate outside the special fibers (see notation from §2.2) π : D x × D y \ (C ∪ L) → D x \ Z n is a locally trivial fibration outside the fibers L whose intersection with C has less than d points.
A classical result by Artin [7] states that the set of geometric bases is in bĳection with Diff + (D \ Z n , ∂D) ∼ = B n .

Remark 3.4.
Due to the fact that any projective plane curve (outside its singular points) is an oriented Riemann surface, the braids obtained in any braid monodromy representation are quasi-positive, that is, they are conjugate of positive braids (braids that can be written as products of positive powers of the standard generators σ i ).
Moreover, the braids that appear in any braid monodromy representation are called algebraic braids because they can be realized as local monodromy of an algebraic function.
Our purpose is to construct an invariant of the projection π. In order to do so, we need to understand the different braid monodromy representations of C relative to (π, Γ, z 0 , s * ) for the different choices of Γ, z 0 , and s * .
(2) Choice of section, or analogously, choice of base point q 0 ∈ {z 0 } × D y = F z 0 . This produces, as mentioned in Remark 1.2, an inner automorphism, that is, a conjugation by a braid β ∈ B d . Hence, there is another action: It is a mere exercise to check that the action of B n and B d on the set of geometric bases commute. This means that there is an right action of B n × B d on the set of monodromy representations, which takes care of all the possible choices of Γ, base points, and sections. Such an action is called the Hurwitz moves of a monodromy representation. Summarizing: Theorem 3.5. Given a monodromy representation µ of C with respect to (π, Γ, z 0 ). There is a one-to-one map between {Monodromy representations of C with respect to π} and { Hurwitz class of µ} .
Definition 3.6. Two monodromy representations of C are called (Hurwitz) equivalent if they belong to the same orbit by the Hurwitz moves described above. That is, if there exists (σ, β) ∈ B n × B d such that µΓ = µΓ (σ,β) . The orbit of a braid monodromy representation by the action of Hurwitz moves will be called the braid monodromy class of a curve. Remark 3.7. Note that µ(γ n )µ(γ n−1 ) · · · µ(γ 2 )µ(γ 1 ) = µ(∂D x ), Since ∂D can be seen as a meridian of the point at infinity of C , that is, the projection of the line C 0 . The condition that C 0 is transversal to C implies that µ∂D = ∆ 2 d = (σ 1 · · · σ d−1 ) d , the Garside element of B d , generator of its center. Thus, µ(γ n )µ(γ n−1 ) · · · µ(γ 2 )µ(γ 1 ) = ∆ 2 d = (σ 1 · · · σ d−1 ) d . This is another way to present a monodromy representation. This is usually known as a Braid Monodromy Factorization.
Many questions are still open regarding braid monodromy factorizations of algebraic curves. We mention just a few:  [56].
The braid monodromy factorization of a smooth curve is a product of conjugates of the standard generators σ i . Any two such products are Hurwitz equivalent [10,Corollary 3.5], therefore, by Theorem 3.5 the realization problem is solved for the smooth case.
One can also find interesting local versions of the realization problem [62].
So far we have proved (Zariski-van Kampen Theorem 2.6) that the braid monodromy representation of a curve determines the fundamental group of its complement. In fact it is much stronger than that, as shows the following result, which has been proved by Kulikov-Teicher in [48] for cuspidal curves and by Carmona [14] in full generality.
Theorem 3.8. The braid monodromy class of C fully determines the topology of the pair (P 2 , C). In other words, if two curves C 1 and C 2 have the same braid monodromy class, then there is a homeomorphism ϕ : The converse is not known in general, basically because the homeomorphism ϕ : P 2 → P 2 may not send lines to lines. Therefore, the pencil of lines through P , which determines the braid monodromy of C 1 is not preserved by ϕ. There are some partial positive converses: Theorem 3.9 (Carmona [14]). The pair (P 2 , C) fully determines the braid monodromy class of C with respect to a projection.
Another partial result in this direction is the following. Let C 1 and C 2 be two curves and L 1 , L 2 be lines such that the affine curves C i ∪ L 1 ⊂ C 2 := P 2 \L i have no vertical asymptotes. Consider L 1 and L 2 the union of vertical lines as described in §2.2 and ϕ : (P 2 , C 1 , L 1 , L 1 ) → (P 2 , C 2 , L 2 , L 2 ) a homeomorphism, then one has the following: Theorem 3.10 (Artal, Carmona,- [2]). The braid monodromy factorization of C 1 from a point P ∈ L 1 is Hurwitz equivalent to the braid monodromy representation of C 2 from ϕ(P ) ∈ L 2 .
In a different direction, there is also a negative converse to Theorem 3.8. Theorem 3.11 ). There are two sequences of plane irreducible cuspidal curves, C m,1 and C m,2 , m ≥ 5, such that the pairs (C 2 , C m,1 ) and (C 2 , C m,2 ) are diffeomorphic, but C m,1 and C m,2 are not isotopic and have different braid monodromy classes.
Obviously, the diffeomorphisms cannot be extended to P 2 , otherwise the hypothesis in Theorem 3.10 would hold and the braid monodromy representations would be equivalent.

The Homotopy Type of (C 2 , C)
Let us consider the affine curve scenario as described at the beginning of §3.1, that is, where D x is big enough to contain all the critical values Z n of the projection from C.
Let us take a closer look at the relations (2.4) in Example 2.20. Note that, the relations derived from the tacnode σ 8 1 (involving branches 1 and 2), are trivial for the generators g 3 and g 4 . Analogously, the braid σ 2 coming from the first tangency (involving branches 2 and 3) preserves generators g 1 and g 4 . This is a general result. To be more precise, let γ i = ω · γ i · ω −1 be a meridian around z i ∈ Z n and let g i 1 , . . . , g i k denote the meridians that approach the singular point x i over z i (see Figure 3.2) when running along ω. The following result is well known, see [50,57].
(1) the 0-dimensional skeleton of the complex will be given by only one 0-cell, the 1-dimensional skeleton of the complex will be in bĳection with the set of generators, say K 1 = {e 1 1 , . . . , e 1 d }, whose boundary will be glued to e 0 , and (3) the 1-dimensional skeleton of the complex will be in bĳection with the set of relators, say K 2 = {e 2 1 , . . . , e 2 n }. The identification morphism is so that ∂e 2 i is glued to the 1-cell r i (ē 1 ).
Remark 3.14. Note that the 2-dimensional CW-complex is associated with a presentation, not with the group. However, certain transformations in the presentation are allowed keeping the homotopy type. Such transformations are called Tietze transformations of type (I) and (II) (cf. [27]): (I) Adding (deleting) a generator g i and a relation of the type g i = w(g 1 , . . . , g i−1 , g i+1 , . . . , g d ).
(II) Replacing a relation r = 1 by a relation r = wsw −1 , where w is any word and s = 1 is another relation.
Tietze transformations of type (III) change the homotopy type of the complex since it means attaching (resp. detaching) a 2-dimensional sphere and this increases (resp. decreases) the Euler characteristic of the complex by one.
Theorem 3.15 (Libgober [50]). The 2-dimensional complex associated with the Zariski presentation has the homotopy type of C 2 \ C.
Proof. The proof of this result is based on two local results. The following is a Zariski presentation (see (2.4)): π 1 (C 2 \C) = g 1 , g 2 , g 3 , g 4 : [(g 2 g 1 ) 4 , g 1 ] = 1, g 2 = g 3 , g 2 = g 3 , g 4 = g 2 g 1 g −1 2 , g 4 = g 2 g 1 g −1 2 ≡ Our focus of attention in this survey will be to describe a simple method to obtain a braid monodromy representation of complexified real arrangements, which will be later extended to other real curves and general line arrangements. Note that both concepts are not equivalent, since MacLane arrangement (see [53]) is real, but not strongly real.
The following result is immediate, but it is worth mentioning for clarity. (2) the plane R 2 ⊂ C 2 is such that R 2 ∩ L is isomorphic to a graph Γ with the following structure (see Figure 3. 3).  Associated with any wiring diagram Γ one can construct a finite list of braids in B d as follows.
Let (x 1 , y 1 ), . . . , (x n , y n ) denote the singular points of Γ ordered such that x 1 > · · · > x n . Denote byδ i := δ i 1 , . . . , δ i k i the segments intersecting at (x i , y i ). One defines β i ∈ B d as strings (1, . . . , k). Globally, one needs to keep track of the position of the segments δ 1 , . . . , δ k .  This gives one a simple algorithmic method to compute braid monodromy representations of complexified strongly real line arrangements.
Finally, note that, despite the simplicity of the method, braid monodromy representations of strongly real arrangements are not determined by the combinatorics. In [3] the authors present two strongly real arrangements of lines with the same combinatorics but whose braid monodromies are not Hurwitz equivalent (see §3.4).
This diagram has two possible generalizations: braided wiring diagrams for complex line arrangements and decorated wiring diagrams for complexified strongly real curves, which will be treated separately.

Braided Wiring Diagrams
Consider a line arrangement L in the affine situation as in §3.1. We recall that the projection π| L has a finite set of critical values denoted by Z n ⊂ D x .
Choose a piecewise linear path starting at a base point on ∂D y with no self-intersections and joining all the points in Z n such that the segment is not broken at the points in Z n . For instance, one can follow the lexicographic order in the complex numbers as in Figure 3.5: The preimage of each segment will be an open braid and not a planar graph as for wiring diagrams. The rest is analogous to the wiring case: when crossing a point in Z n , where the linesδ = (δ 1 , . . . , δ k ) converge, a braid of local type ∆δ will be generated and when ending at a point in Z n a braid of local type ∆ 2 δ will appear.   Recall that f 1 := 4 5 9 and f 2 := 1 8 10 generate a pencil of cubics containing f 3 := 2 7 11 and f 4 := 3 6 12 as members of the pencil. Note that each reducible cubic f 1 , . . . , f 4 consists of three lines in general position (three lines joining all the inflexion points of a smooth cubic, each one containing three of them). Using the results in [47] on braid factorizations, one can prove that the Hesse arrangement cannot degenerate onto a pseudoholomorphic Hesse arrangement, where the cubics f i become three concurrent lines.

Decorated Wiring Diagrams
Definition 3.25. A plane curve C = C 1 · · · C r is called real if there is a projective system of coordinates such that C admits real equations. If, in addition, C 1 , . . . , C r admit real equations, the singular points and vertical tangencies have real coordinates, and the tangent cone at each singularity is a strongly real line arrangement, we call C a strongly real curve.
Consider the affine situation of a strongly real curve of degree d and the vertical projection (onto the first coordinate) as in §3.1. One can write a diagram with solid lines and dashed lines, where:   Note, for example, notice the braids generated at the local picture:

Conjugated Curves
One last application of braid monodromies is the study of the different Galois embeddings of a curve given by equations in a number field in the spirit of [71,1]. More precisely, let C be a plane curve whose equation can be defined on the ring of polynomials with coefficients on a number field K ⊃ Q. Any Galois transformation σ of the number field will produce another curve C σ , whose equation is again defined on K[X, Y, Z], with the same number and degrees of irreducible components, same type of singularities,... that is same combinatorial type. However, since σ cannot necessarily be extended to a homeomorphism of the total space (think of the automorphism √ 2 → − √ 2) the question arises whether or not (P 2 , C) and (P 2 , C σ ) are topologically equivalent.
Also note that any such example cannot be detected by means of algebraic invariants such as the algebraic fundamental group (that is, the profinite completion of the fundamental group), Alexander polynomials, or any kind of invariant related with finite coverings.
In our particular example, one can use the Burau representation of B 5 into GL(5; Z[t ±1 ]). Replacing t by 2 mod 5 one obtains a representation β : B 5 → GL(5; Z/5Z) such that the braid monodromy representations produce different orbits after the Hurwitz action.
Therefore, using Theorem 3.10 one obtains the following.
Remark 3.28. Still, the question whether or not the fundamental groups G + := π 1 (P 2 \ C + ) and G − := π 1 (P 2 \ C − ) are isomorphic remains open. As mentioned above, the reader should notice that π alg 1 (P 2 \ C + ) ∼ = π alg 1 (P 2 \ C − ). In other words, G + and G − have the same profinite completion (that is, the same structure of finite index subgroups). This is a paradigmatic example in the sense that it shows the power of the braid monodromy representation of a curve in and of itself and not as a mere instrument to obtain fundamental groups.