On the connection between fundamental groups and pencils with multiple fibers

We present two results about the relationship between fundamental groups of quasiprojective manifolds and linear systems on a projectivization. We prove the existence of a plane curve with non-abelian fundamental group of the complement which does not admit a mapping onto an orbifold with non-abelian fundamental group. We also find an affine manifold whose irreducible components of its characteristic varieties do not come from the pull-back of the characteristic varieties of an orbifold

On the topology of fibered-type curves: a Zariski pair of affine nodal curves

In this paper we explore conditions for the complement of a fibered-type curve to be a free product of cyclic groups. We exhibit a Zariski pair of curves in $\mathbb{C}^2$ with only nodes as singularities (and the same singularities at infinity) whose complements have non-isomorphic fundamental groups, one of them being free. This shows that the position of singularities also affects this problem and hence the sufficient conditions involve more than local invariants. We also study the homotopy type of projective curve complements whose fundamental groups are free products of cyclic groups. Finally we describe the CW-complex structure of certain fibered-type curve complements.Comment: 14 pages. Comments are welcome and greatly appreciate

Modelos, axiomática y geometría del plano hiperbólico

First of all, we need to understand why there are other geometries such as Hyperbolic geometry besides the intuitive Euclidean geometry. Who discovered this geometry? Lobachevski, a young scientist who decided to leave the medical career to devote himself completely to the study of a geometry that he called “imaginary geometry”, is a founder of this geometry. He made progress not only in mathematics but also in physics, such as Einstein’s theory of relativity. How did Lobachevski come up with this geometry? Before giving an answer to this question, let’s see what an axiomatic system is

On the Sigma-invariants of even Artin groups of FC-type

In this paper we study Sigma-invariants of even Artin groups of FC-type, extending some known results for right-angled Artin groups. In particular, we define a condition that we call the strong n-link condition for a graph Gamma and prove that it gives a sufficient condition for a character chi : A(Gamma) -> Z to satisfy chi] is an element of Sigma(n)(A(Gamma), Z). This implies that the kernel A(Gamma)(chi) = ker chi is of type FPn. We prove the homotopical version of this result as well and discuss partial results on the converse. We also provide a general formula for the free part of H-n(A(Gamma)(chi); F) as an Ft(+/- 1)]-module with the natural action induced by chi. This gives a characterization of when H-n(A(Gamma)(chi); F) is a finite dimensional vector space over F

John Willard Milnor, 1962 Fields Medal. (Spanish: John Willard Milnor, Medalla Fields 1962)

Depto. de Álgebra, Geometría y TopologíaFac. de Ciencias MatemáticasFALSEpu

Triangular curves and cyclotomic Zariski tuples

The purpose of this paper is to exhibit infinite families of conjugate projective curves in a number field whose complement have the same abelian fundamental group, but are non-homeomorphic. In particular, for any d=4 we find Zariski (¿d2¿+1)-tuples parametrized by the d-roots of unity up to complex conjugation. As a consequence, for any divisor m of d, m¿1,2,3,4,6, we find arithmetic Zariski ¿(m)2-tuples with coefficients in the corresponding cyclotomic field. These curves have abelian fundamental group and they are distinguished using a linking invariant

Cyclic coverings of rational normal surfaces which are quotients of a product of curves

This paper deals with cyclic covers of a large family of rational normal surfaces that can also be described as quotients of a product, where the factors are cyclic covers of algebraic curves. We use a generalization of Esnault-Viehweg method to show that the action of the monodromy on the first Betti group of the covering (and its Hodge structure) splits as a direct sum of the same data for some specific cyclic covers over $\mathbb{P}^1$. This has applications to the study of L\^e-Yomdin surface singularities, in particular to the action of the monodromy on the Mixed Hodge Structure, as well as to isotrivial fibered surfaces.Comment: Accepted in Publicacions Matem\`atique