Abstract. We show that the degree of the Alexander polynomial of an irreducible plane algebraic curve with nodes and cusps as the only singularities does not exceedd−2 where d is the degree of the curve. We also show that the Alexander polynomial ∆ C (t) of an irreducible curve C = {F = 0} ⊂ P 2 whose singularities are nodes and cusps is non-trivial if and only if there exist homogeneous polynomials f , g, and h such that f 3 +g 2 +F h 6 = 0. This is obtained as a consequence of the correspondence, described here, between Alexander polynomials and ranks of Mordell-Weil groups of certain threefolds over function fields. All results also are extended to the case of reducible curves and Alexander polynomials ∆ C,ε (t) corresponding to surjections ε : π 1 (P 2 \ C 0 ∪ C) → Z, where C 0 is a line at infinity. In addition, we provide a detailed description of the collection of relations of F as above in terms of the multiplicities of the roots of ∆ C,ε (t). This generalization is made in the context of a larger class of singularities i.e. those which lead to rational orbifolds of elliptic type.

The paper studies a relation between fundamental group of the complement to a plane singular curve and the orbifold pencils containing it. The main tool is the use of Albanese varieties of cyclic covers ramified along such curves. Our results give sufficient conditions for a plane singular curve to belong to an orbifold pencil, i.e. a pencil of plane curves with multiple fibers inducing a map onto an orbifold curve whose orbifold fundamental group is non trivial. We construct an example of a cyclic cover of the projective plane which is an abelian surface isomorphic to the Jacobian of a curve of genus 2 illustrating the extent to which these conditions are necessary.

In this work, we study a family of Cremona transformations of weighted projective planes which generalize the standard Cremona transformation of the projective plane. Starting from special plane projective curves we construct families of curves in weighted projective planes with special properties. We explain how to compute the fundamental groups of their complements, using the blow-up-down decompositions of the Cremona transformations, we find examples of Zariski pairs in weighted projective planes (distinguished by the Alexander polynomial). As another application of this machinery we study a family of singularities called weighted Lê-Yomdin, which provide infinite examples of surface singularities with a rational homology sphere link. To end this paper we also study a family of surface singularities generalizing Brieskorn-Pham singularities in a different direction. This family contains infinitely many new examples of integral homology sphere links, answering a question by Némethi.

No abstract

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 K[t ±1 ], where 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.