The Max Noether Fundamental Theorem is Combinatorial

In the present paper we give a reformulation of the Noether Fundamental Theorem for the special case where the three curves involved have the same degree. In this reformulation, the local Noether's Conditions are weakened. To do so we introduce the concept of Abstract Curve Combinatorics (ACC) which will be, in the context of plane curves, the analogue of matroids for hyperplane arrangements

Characteristic varieties of graph manifolds and quasi-projectivity of fundamental groups of algebraic links

The present paper studies the structure of characteristic varieties of fundamental groups of graph manifolds. As a consequence, a simple proof of Papadima's question is provided on the characterization of algebraic links that have quasi-projective fundamental groups. The type of quasi-projective obstructions used here are in the spirit of Papadima's original work.Comment: 22 pages, 6 figures, to appear in European Journal of Mathematic

Module structure of the homology of right-angled Artin kernels

In this paper, we study the module structure of the homology of Artin kernels, i.e., kernels of non-resonant characters from right-angled Artin groups onto the integer numbers, the module structure being with respect to the ring $\mathbb{K}[t^{\pm 1}]$, where $\mathbb{K}$ is a field of characteristic zero. Papadima and Suciu determined some part of this structure by means of the flag complex of the graph of the Artin group. In this work, we provide more properties of the torsion part of this module, e.g., the dimension of each primary part and the maximal size of Jordan forms (if we interpret the torsion structure in terms of a linear map). These properties are stated in terms of homology properties of suitable filtrations of the flag complex and suitable double covers of an associated toric complex.Comment: 24 pages, 6 figure

An arithmetic Zariski pair of line arrangements with non-isomorphic fundamental group

In a previous work, the third named author found a combinatorics of line arrangements whose realizations live in the cyclotomic group of the fifth roots of unity and such that their non-complex-conjugate embedding are not topologically equivalent in the sense that they are not embedded in the same way in the complex projective plane. That work does not imply that the complements of the arrangements are not homeomorphic. In this work we prove that the fundamental groups of the complements are not isomorphic. It provides the first example of a pair of Galois-conjugate plane curves such that the fundamental groups of their complements are not isomorphic (despite the fact that they have isomorphic profinite completions)