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

Abstract

In this paper we investigate Uludag's method for constructing new curves whose fundamental groups are central extensions of the fundamental group of the original curve by finite cyclic groups.In the first part, we give some generalizations to his method in order to get new families of curves with controlled fundamental groups. In the second part, we discuss some properties of groups which are preserved by these methods. Afterwards, we describe precisely the families of curves which can be obtained by applying the generalized methods to several types of plane curves. We also give an application of the general methods for constructing new Zariski pairs.

On the fundamental group of the complement of two real tangent conics and an arbitrary number of real tangent lines

“…In a similar way, we get [x i , x n+2 x n x −1 n+2 ] = e, for 1 ≤ i ≤ n − 3. As done above, we substitute this time Relation (10) in Relation (9) to get: [x n−1 , x n+2 x n x −1 n+2 ] = e. We combine both induced relations in Relations (5) below, to get:…”

“…By Relations (10), Relations (9) become [x 4 , x i ] = e, where 6 ≤ i ≤ n + 4. These relations enable us to proceed in simplification:…”

“…We start by substituting i = 6 in Relations (6) to get (x 2 x 6 ) 2 = (x 6 x 2 ) 2 . We continue with i = 6, j = 7 in Relations (10) to get [x −1 2 x 6 x 2 , x 7 ] = e (using (x 2 x 6 ) 2 = (x 6 x 2 ) 2 ). Using these two relations, we simplify Relations (6) for i = 7: (x 2 x −1 6 x 7 x 6 ) 2 = (x −1 6 x 7 x 6 x 2 ) 2 .…”

“…the same singular points and the same arrangement of irreducible components), but their complements are not homeomorphic. Some examples of Zariski pairs can be found at [6], [7], [9], [10], [21], [22], [23] and [24].…”

Knot Theory of Complex Plane Curves

“…The background and motivation for the conjecture, and related material on knotgroups of complex plane curves, may be found in Enriques (1923); Lefschetz (1924);Brauner (1928);Zariski (1929);van Kampen (1933); Burau (1934); Zariski (1936Zariski ( , 1971; Reeve (1954Reeve ( /1955Cheniot (1973);Lê Dũng Tráng (1974);Oka (1975Oka ( /76, 1974; ; Chang (1979);Randell (1980); Kaliman (1992); Moishezon and Teicher (1996); Kulikov (1997);Garber (2003).…”