Abstract. In this paper we show a Zariski pair of sextics which is not a degeneration of the original example given by Zariski. This is the first example of this kind known. The two curves of the pair have a trivial Alexander polynomial. The difference in the topology of their complements can only be detected via finer invariants or techniques. In our case the generic braid monodromies, the fundamental groups, the characteristic varieties and the existence of dihedral coverings of P 2 ramified along them can be used to distinguish the two sextics. Our intention is not only to use different methods and give a general description of them but also to bring together different perspectives of the same problem.
We study the fundamental groups of the complements to curves on simply connected surfaces, admitting non-abelian free groups as their quotients. We show that given a subset of the Néron–Severi group of such a surface, there are only finitely many classes of equisingular isotopy of curves with irreducible components belonging to this subset for which the fundamental groups of the complement admit surjections onto a free group of a given sufficiently large rank. Examples of subsets of the Néron–Severi group are given with infinitely many isotopy classes of curves with irreducible components from such a subset and fundamental groups of the complements admitting surjections on a free group only of a small rank.
In [1], Hirano gives a method for constructing families of curves with a large number of singularities. The idea is to consider an abelian covering of P2 ramified along three lines in general position, and to take the pull‐back of a curve C intersecting the lines non‐generically. Similar constructions are used by Shimada in [10] and Oka in [8]. We apply this method for the case where C is a conic, constructing a family of curves with the following asymptotic behaviour (see [9]):
limn→∞(∑p∈Sing C˜nμ(p))/(deg(C˜n))2=34.
The goal of this paper is to calculate the fundamental group for the curves in this family as well as their Alexander polynomial. 1991 Mathematics Subject Classification 14H20, 14H30, 14E20.
Abstract. In this work we study the connection between the existence of finite dihedral covers of the projective plane ramified along an algebraic curve C, infinite dihedral covers, and pencils of curves containing C.
We consider the structure of reducible curves on a projective simply connected surface with irreducible components belonging to a selected subset the effective cone of the surface and which fundamental groups of the complements admit free quotients having rank greater than one. Main result is the following trichotomy depending on the ranks of free (essential) quotients of the fundamental groups with components in the subset of effective cone. A: There can be an infinite number of isotopy classes of curves with classes of components in a selected subset of effective cone and rank of free quotients being below a threshold depending on the subset. B: There are only finitely many isotopy classes of curves with components in selected subset of effective cone admitting surjection onto a free group of rank greater the threshold. C: Moreover, irreducible components of curves admitting an essential surjection onto a free group of rank sufficiently larger than the threshold belong to a pencil of curves having class in the selected subset of the effective cone. Some explicit information on the thresholds for different cases of the trichotomy are discussed.
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