Jorge Carmona Ruber

Braid monodromy and topology of plane curves

In this paper we prove that braid monodromy of an affine plane curve determines the topology of a related projective plane curve. Introduction Our purpose in this paper is to relate the topological embedding of algebraic curves to a refinement of a well-known invariant of curves such as braid monodromy. Roughly speaking, braid monodromy is defined for a triple (C , L , P), whereC ⊂ P2 is a curve,L ⊂ P2 is a line not contained inC , andP ∈ L, as follows. Let us consider homogeneous coordinates [x : y : z] such thatP = [0 : 1 : 0], L = {z = 0}, and C = { f (x, y, z) = 0}. Let d be they-degree ofC ; that is,d = degy( f (x, y, 1)). The pencilH of lines passing through P (and different fromL) is parametrized byx ∈ C. By the theorem of continuity of roots, H determines a representation of a free group F on the braid group ond strings, which is called a braid monodromy of the triple (C , L , P). The free groupF corresponds to the fundamental group of an r -punctured complex line, where the punctures come from the nongeneric elements of H with respect toC . The classical definition of braid monodromy refers to generic choices of L and P, for example,P / ∈ C andL transversal toC . In this work we allow certain nongeneric choices. Braid monodromy is a strong invariant of plane curves. It is fair to say that the main ideas that lead to this invariant have already been used in the classic works of O. Zariski [13] and E. van Kampen [ 6] to find the fundamental group of the complement of a curve. The first explicit definition of braid monodromy was made by DUKE MATHEMATICAL JOURNAL Vol. 118, No. 2, c

Topology and combinatorics of real line arrangements.

We prove the existence of complexified real arrangements with the same combinatorics but different embeddings in the complex projective plane. Such pair of arrangements has an additional property: they admit conjugated equations on the ring of polynomials over the number field

On sextic curves with big Milnor number.

{\mathbb Q}(\sqrt{5})

Effective invariants of braid monodromy

.We prove the existence of complexified real arrangements with the same combinatorics but different embeddings in the complex projective plane. Such pair of arrangements has an additional property: they admit conjugated equations on the ring of polynomials over the number field

Zariski pairs, fundamental groups and Alexander polynomials

{\mathbb Q}(\sqrt{5})

Invariants of combinatorial line arrangements and Rybnikov's example

.

Fundamental group of plane curves and related invariants

In this work we present an exhaustive description, up to projective isomorphism, of all irreducible sextic curves in ℙ2 having a singular point of type ,\( \mathbb{A}_n ,n \geqslant 15 \) n ≥ 15, only rational singularities and global Milnor number at least 18. Moreover, we develop a method for an explicit construction of sextic curves with at least eight — possibly infinitely near — double points. This method allows us to express such sextic curves in terms of arrangements of curves with lower degrees and it provides a geometric picture of possible deformations. Because of the large number of cases, we have chosen to carry out only a few to give some insights into the general situation.

Topological and arithmetical properties of rational plane curves

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.In this paper we construct effective invariants for braid monodromy of affine curves. We also prove that, for some curves, braid monodromy determines their topology. We apply this result to find a pair of curves with conjugate equations in a number field but which do not admit any orientation-preserving homeomorphism.