Effective invariants of braid monodromy

Abstract

Abstract. In this paper we construct new invariants of algebraic curves based on (not necessarily generic) braid monodromies. Such invariants are effective in the sense that their computation allows for the study of Zariski pairs of plane curves. Moreover, the Zariski pairs found in this work correspond to curves having conjugate equations in a number field, and hence are not distinguishable by means of computing algebraic coverings. We prove that the embeddings of the curves in the plane are not homeomorphic. We also apply these results to the classification problem of elliptic surfaces.Our purpose in this paper is the construction of new and effective topological invariants for algebraic curves. These invariants will be derived from the braid monodromy of a triple (C, L, P ) introduced in [3], where C ⊂ P 2 is an algebraic curve, L ⊂ C is a line and P ∈ L. This invariant is derived from the classical braid monodromy; see references in [3].Several topological invariants of plane algebraic curves, obtained from the fundamental group of the complement, have been found to be effective, such as the Alexander polynomial, the Alexander module or the sequence of characteristic varieties [13]. A second group of effective invariants, described by A. Libgober in [12], depend on polynomial representations of the braid group and are invariants of the conjugation class of the monodromy group.In this paper, braid monodromy of a triple is represented by an element of B Using finite representations of braid groups we obtain new effective invariants for braid monodromies. Effectiveness is obtained by means of the free software GAP4 [10]. We combine these invariants with our main theorem in [3] which relates braid monodromy to topology.Braid monodromy may also be useful in the study of the moduli space of projective curves with prescribed degree and topological types of curve singularities. Let d be a positive integer and let T 1 , . . . , T r be topological types of curve singularities. Let us denote by Σ(T 1 , . . . , T r ; d) the space of all projective plane curves of degree d with exactly r singular points of type T 1 , . . . , T r . Let M(T 1 , . . . , T r ; d) be the

Criterion of Hurwitz equivalence for quasipositive factorizations of 3-braids

“…The problem of Hurwitz equivalence of n-tuples of braids appears in the study of braid monodromy of algebraic curves in C 2 . It was considered by many authors, see [1,3,4] and references in [3]. We give an answer for the case mentioned in the title.…”

Zariski $k$-plets via dessins d'enfants

“…Here, the domain of j is the base of the ruling Σ k → P 1 , whereas its range is the standard projective line P 1 = C 1 ∪ {∞}. If the fiber F z over z ∈ P 1 is nonsingular, then the value j(z) is the usual j-invariant (divided by the magic number 1728 = 12 3 ) of the quadruple of points cut on F z by the union B ∪ E (or, in more conventional terms, the j-invariant of the elliptic curve that is the double of F z ∼ = P 1 ramified at the four points above). The values of j at the finitely many remaining points corresponding to the singular fibers of B are obtained by analytic continuation.…”

“…Since l(∞) = 0, 1, one has deg p = deg q. For each root a of p of multiplicity 1 mod 3 (respectively, 2 mod 3), multiply both p and q by (z − a) 2 (respectively, (z − a) 4 ), and for each root b of q of multiplicity 1 mod 2, multiply both p and q by (z − b) 3 . In the resulting representation l = p/q, the multiplicity of each root of p (respectively, q) is divisible by 3 (respectively, 2), and p and q have no common roots of multiplicity 6.…”

“…In the resulting representation l = p/q, the multiplicity of each root of p (respectively, q) is divisible by 3 (respectively, 2), and p and q have no common roots of multiplicity 6. Hence, one has p = 4g 3 2 and q = 27g 2 3 for some polynomials g 2 , g 3 satisfying the condition in Proposition 3.1.1, and the function j = l/(l + 1) = p/(p + q) is the j-invariant of the simplified trigonal curve B ⊂ Σ k given by the Weierstraß equation with coefficients g 2 , g 3 , where k = 1 6 deg p = 1 6 deg q. Clearly, the polynomials g 2 , g 3 as above are defined by l uniquely up to the transformation given by (3.1).…”

A note on the topology of the complements of fiber-type line arrangements in ℂℙ2

“…To study such problems, mathematicians have developed many techniques, for example, the lattice-isotopy theorem and braid monodromy method which will be used in this paper. The lattice-isotopy theorem was used in [Jiang and Yau 1994;Wang and Yau 2005;2007;2008;Yau and Ye 2009] to derive the structures of so-called nice arrangements and prove that their differential structures are determined by their combinatorics. Braid monodromy method has been widely used to study the topology of complements of plane algebraic curves and line arrangements; see, for example, [Moishezon 1981;Cohen and Suciu 1997;Dung 1999;Kulikov and Taȋkher 2000;Cohen 2001; Artal Bartolo et al 2003;2007].…”