Fomento del software libre en los estudios de matemáticas mediante el uso de SAGE.

Fac. de Ciencias MatemáticasFALSEsubmitte

Essential coordinate components of characteristic varieties

In this note we give an algebraic and topological interpretation of essential coordinate components of characteristic varieties and illustrate their importance with an example.Comment: 15 pages, 4 postscript figures, to appear in Mathematical Proceedings of the Cambridge Philosophical Societ

Invariants of combinatorial line arrangements and Rybnikov's example

Following the general strategy proposed by G.Rybnikov, we present a proof of his well-known result, that is, the existence of two arrangements of lines having the same combinatorial type, but nonisomorphic fundamental groups. To do so, the Alexander Invariant and certain invariants of combinatorial line arrangements are presented and developed for combinatorics with only double and triple points. This is part of a more general project to better understand the relationship between topology and combinatorics of line arrangements

Fundamental group of plane curves and related invariants

The article under review contains a study of the topology of a pair (P2,C), where C is an algebraic curve in the complex projective plane. The basic problem is to find invariants which are sensitive enough to distinguish many pairs, and for which there is an algorithm for checking this. The homology of the complement is certainly computable in this sense, but it is too coarse to be really useful. The fundamental group of the complement, by contrast, is very sensitive. The article reviews the Zariski-van Kampen method for finding a presentation for it. However, it is not clear whether the isomorphism problem for this class of groups is solvable. The article surveys many other invariants, such as the Alexander polynomial and characteristic varieties, which are more computable. This last set of invariants was introduced, in this context, by A. S. Libgober [in Applications of algebraic geometry to coding theory, physics and computation (Eilat, 2001), 215–254, Kluwer Acad. Publ., Dordrecht, 200

Tutoriales interactivos para el estudio de la programación: impacto en el aprendizaje

Depto. de Sistemas Informáticos y ComputaciónFac. de InformáticaFALSEsubmitte

Zariski pairs, fundamental groups and Alexander polynomials.

Sección Deptal. de Sistemas Informáticos y ComputaciónFac. de Ciencias MatemáticasTRUECAICYTpu

Topology and combinatorics of real line arrangements.

We prove the existence of complexified real arrangements with the same combinatorics but different embeddings in P2. Such a pair of arrangements has an additional property: they admit conjugated equations on the ring of polynomials over Q(√5)

On a conjecture by A. Durfee

This note provides a negative answer to the following question of A. H. Durfee [Invent. Math. 28 (1975), 231–241; ]: Is it true for arbitrary polynomials F(x,y,z) having an isolated singularity at the origin that the local monodromy is of finite order if and only if a resolution of F(x,y,z)=0 contains no cycles? Here "the monodromy'' means the action on the cohomology of the Milnor fiber of F corresponding to the degeneration F(x,y,z)=t. The authors consider the following example: F(x,y,z)=(xz−y2)3−((y−x)x2)2+z6. They calculate the graph of the resolution (which is a tree) and invariant polynomials of the monodromy (showing the presence of Jordan blocks of a size greater than 1). The key point in these calculations is that this singularity belongs to the class of superisolated (SIS) surface singularities which was studied in detail by the first named author [Mem. Amer. Math. Soc. 109 (1994), no. 525, x+84 pp.;]. SISs are the singularities of the form F(x,y,z)=f(x,y,z)+lN, where l is a generic linear form, N is a sufficiently large integer and f(x,y,z)=0 is a projective plane algebraic curve, the cone over which is the tangent cone of the singularity F(x,y,z). The main step in detecting that the order of the monodromy of a SIS is infinite is the calculation of the Alexander polynomial [A. S. Libgober, Duke Math. J. 49 (1982), no. 4, 833–851;] of the plane curve f(x,y,z)=0. In the authors' example, the plane sextic (xz−y2)3−((y−x)x2)2 has two singularities with local types u3=v10 and u2=v3 respectively and has as its Alexander polynomial t2−t+1. The latter yields that the monodromy of F has an infinite order. The paper is concluded with a series of other interesting observations on the relation between the topology of resolution and monodromy of SIS singularities