Computational Aspects of Line and Toric Arrangements

In the first part of this thesis we deal with the theory of hyperplane arrangements, that are (finite) collections of hyperplanes in a (finite-dimensional) vector space. If A is an arrangement, the main topological object associated with it is its complement M(A), and the main combinatorial object is its intersection poset L(A). Many studies have been done in order to understand what topological properties of M(A) can be inferred from the knowledge of the combinatorial data of L(A). In the thesis we focus on the characteristic variety V(A) associated with an arrangement A, in an effort to uncover some possible combinatorial description of it. Characteristic varieties have been studied for some years and information about them could shed more light on other topological objects, such as the so-called Milnor fibre of an arrangement. Unfortunately there are not many examples of computed characteristic varieties in the literature, because the algorithms involved require many computational resources and are both time- and memory-consuming. To overcome this problem, we developed some new algorithms that are able to compute characteristic varieties for general arrangements. In Chapter 4 we describe them in full details, together with actual SageMath code, so that other researchers could follow our path; in Chapter 5 we collect the results in a little catalogue. We tried to evince some general combinatorial pattern from them, but we leave our considerations in conjecture form. The second part of the thesis is focused on toric arrangements, which are finite collections of subtori (called layers in this context) in the complex algebraic torus. In particular, we follow two articles by De Concini and Gaiffi in which they compute projective wonderful models for the complement of a toric arrangement and a presentation of their integer cohomology rings. Also in this case we develop an algorithm that is able to produce examples of such rings. This is a little step, but we hope that the possibility to compute more examples, together with better and more efficient algorithms, can greatly improve the understanding of these topological objects.