More Zariski pairs and finite fundamental groups of curve complements

On Some Conjectures About Free and Nearly Free Divisors

“…Kummer covers are a very useful tool in order to construct complicated algebraic curves starting from simple ones. Since Kummer covers are finite Galois unramified covers of P 2 \ {xyz = 0} with Gal(π k ) ∼ = Z/kZ × Z/kZ, topological properties of the new curves can be obtained: Alexander polynomial, fundamental group, characteristic varieties and so on (see [1,3,5,28,17,6,4,18] for papers using these techniques). Example 3.2.…”

“…In[28], Uludag constructs new examples of Zariski pairs using former ones and Kummer covers. He also uses the same techniques to construct infinite families of curves with finite non-abelian fundamental groups.…”

Plane curves and their fundamental groups: Generalizations of Uludağ’s construction

“…We first recall the definition of a meridian of a curve C at a point p (see [32,33]): Let ∆ be a smooth analytical branch meeting C transversally at p and let x 0 ∈ P 2 − C be a base point. Take a path ω joining x 0 to a boundary point of ∆, and define the meridian of C at p to be the loop µ p = ω · δ · ω −1 , where δ is the boundary of ∆, oriented in the positive sense (see Figure 3).…”

Kummer covers and braid monodromy

“…Since Kummer covers are finite Galois covers of P 2 \ {xyz = 0} with Gal(π n ) ∼ = Z/nZ × Z/nZ, topological properties of the new curves can be obtained: Alexander polynomial, fundamental group, characteristic varieties, and so on (see [3,4,17,41,25,19] for papers using these techniques). Kummer covers are a very useful tool in order to construct complicated algebraic curves starting from simple ones.…”

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In this work, we describe a method to construct the generic braid monodromy of the preimage of a curve by a Kummer cover. Then (∇, σ x , σ y ) determines the generic braid monodromy of C n .The fundamental group of the complement of a curve under a generic Kummer cover was computed by Uludag [41]. The latter process, called generification, is independent from Kummer covers, and it can be applied in more general circumstances since non-generic braid monodromies appear more naturally and are oftentimes much easier to compute.

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