Let C be a reduced curve of degree d in the complex projective plane, defined by a homogeneous polynomial f. The monodromy action on the Milnor fiber F:={f=1} of C is given by the multiplication with a primitive root of the unity of order d, and induces a monodromy operator at the level of the cohomology of F.
In this talk we will focus on curve arrangements of pencil type - i.e., those where f is a product of (more than two) elements of a pencil of curves. Although the situation when the curves in the pencil are line arrangements has been systematically considered, in the general case different situations may occur and many questions stay open. I will recall some basic properties of plane curves, with emphasis on eigenvalues of the monodromy and freeness. I will also investigate the differences between the general case and the line arrangement case by giving specific examples.

The study of hyperplane arrangements naturally leads to the study of Coxeter groups, Coxeter arrangements and Artin groups. Many properties, conjectured to be true for all Artin groups, have been proved only for some families. In particular the K(π,1) conjecture says that Coxeter arrangements are Eilenberg-MacLane spaces. We will introduce Coxeter groups and Vinberg systems in order to provide a detailed presentation of the conjecture, with an overview of the cases for which it has been proved. We will define Salvetti complex of a (possibly infinite) arrangement of hyperplanes and sketch a proof of Deligne's theorem. If times allows, we will give also a proof of the conjecture in other cases.

TBA

The starting point of the theory of hyperplane arrangements is the computation of the cohomology ring of the complement of an arrangement. The aim of this talk will be to explain this classical story, started by the work of Arnol’d on the braid group and the work of Brieskorn and Orlik—Solomon. It involves some beautiful interactions between topology, geometry, algebra and combinatorics. If time allows, we will also discuss the more general case of hypersurface arrangements.

Let us consider, in a real or complex vector space V, a subspace arrangement A. We will describe the De Concini-Procesi models associated with A, that are smooth varieties where the union of the subspaces is replaced by a divisor with normal crossings (for instance, if A is the braid hyperplane arrangement, the minimal complex De Concini-Procesi model associated to it is the moduli space of stable genus 0 curves with n + 1 points). We will point out the combinatorial properties of these models, including their connection with the construction of special polytopes (nestohedra); when A is the hyperplane arrangement associated to a reflection group W, we will describe the corresponding group action on cohomology. We will also show how to obtain compact wonderful models in the case of toric arrangements (joint work with Corrado De Concini).

One of the most fruitful ideas to arise from arrangement theory is that of turning an algebraic model for a space into a family of cochain complexes, parametrized by the cohomology group in degree 1, and extracting certain varieties from these data, as the loci where the cohomology of those cochain complexes jumps. What makes these resonance varieties really useful is their close connection with the characteristic varieties, which record the jumps in homology with coefficients in rank 1 local systems. These two sets of jump loci are even more tightly related under certain algebraic (positivity of weights), topological (formality), or geometric (quasi-projectivity) assumptions. Furthermore, in favorable circumstances, the resonance varieties control the Chen ranks of the fundamental group of our space.

Topics: * Resonance varieties of commutative (differential) graded algebras. * Product formulas. * Positive weights and linearity. * Tangent cone theorems. * The Chen ranks formula and possible generalizations. * Examples and applications.