Simona Settepanella

Combinatorial Morse theory and minimality of hyperplane arrangements

which is generic with respect to the arrangement, ie Vi intersects transversally all codimension‐i intersections of hyperplanes. The interesting main result is a correspondence between the k ‐cells of the minimal complex and the set of chambers which intersect Vk but do not intersect Vk 1 . The arguments still use the Morse theoretic proof of the Lefschetz theorem, and some analysis of the critical cells is given. Unfortunately, the description does not allow one to understand exactly the attaching maps of the cells of a minimal complex.

Production Theory: Accounting for Firm Heterogeneity and Technical Change

The paper presents a new framework to assess firm level heterogeneity and to study the rate and direction of technical change. Building on the analysis of revealed short-run production functions by Hildenbrand (1981), we propose the (normalized) volume of the zonotope composed by vectors-firms in a narrowly defined industry as an indicator of inter-firm heterogeneity. Moreover, the angles that the main diagonal of the zonotope form with the axes provide a measure of the rates and directions of technical change over time. The proposed framework can easily account for n-inputs and m-outputs and, crucially, the measures of heterogeneity and technical change do not require many of the standard assumptions from production theory. (This abstract was borrowed from another version of this item.)

VANISHING RESULTS FOR THE COHOMOLOGY OF COMPLEX TORIC HYPERPLANE COMPLEMENTS

Suppose R is the complement of an essential arrangement of toric hyperlanes in the complex torus and ? = ?1(R). We show that H*(R;A) vanishes except in the top degree n when A is one of the following systems of local coefficients: (a) a system of nonresonant coefficients in a complex line bundle, (b) the von Neumann algebra N?, or (c) the group ring Z?. In case (a) the dimension of Hn is the Euler characteristic, e(R), and in case (b) the nth l2 Betti number is also |e(R)|.

Combinatorial polar orderings and recursively orderable arrangements

Polar orderings arose in recent work of Salvetti and the second author on minimal CW-complexes for complexified hyperplane arrangements. We study the combinatorics of these orderings in the classical framework of oriented matroids, and reach thereby a weakening of the conditions required to actually determine such orderings. A class of arrangements for which the construction of the minimal complex is particularly easy, called recursively orderable arrangements, can therefore be combinatorially defined. We initiate the study of this class, giving a complete characterization in dimension 2 and proving that every supersolvable complexified arrangement is recursively orderable.

Homology graph of real arrangements and monodromy of Milnor fiber

We study the first homology group of the Milnor fiber of sharp arrangements in the real projective plane. Our work relies on the minimal Salvetti complex of the deconing arrangement and its boundary map. We describe an algorithm which computes possible eigenvalues of the first monodromy operator. We prove that, if a condition on some intersection points of lines is satisfied, then the only possible non trivial eigenvalues are cubic roots of the unity. Moreover we give sufficient conditions for just eigenvalues of order 3 or 4 to appear in cases in which this condition is not satisfied.

Social choice among complex objects

We present a geometric model of social choice when the latter takes place among bundles of interdependent elements, that we will call objects. We show that the outcome of the social choice process is highly dependent on the way these bundles are formed. By bundling and unbundling the same set of constituent elements an authority has the power of determine the social outcome. We provide necessary and sufficient conditions under which a social outcome may be a local or global optimum for a set of objects, and we show that, by appropriately redefining the set of objects, intransitive cycles may be broken and the median voter may be turned into a loser.

Production theory: accounting for firm heterogeneity and technical change

The paper presents a new framework to assess firm level heterogeneity and to study the rate and direction of technical change. Building on the analysis of revealed short-run production functions by Hildenbrand (1981), we propose the (normalized) volume of the zonotope composed by vectors-firms in a narrowly defined industry as an indicator of inter-firm heterogeneity. Moreover, the angles that the main diagonal of the zonotope form with the axes provide a measure of the rates and directions of technical change over time. The proposed framework can easily account for n-inputs and m-outputs and, crucially, the measures of heterogeneity and technical change do not require many of the standard assumptions from production theory.

Optimality in Social Choice

Marengo and Settepanella (2010) have developed a geometric model of social choice when it takes place among bundles of interdependent elements, showing that by bundling and unbundling the same set of constituent elements an authority has the power of determining the social outcome. In this article, we will tie the model above to tournament theory, solving some of the mathematical problems arising in their work and opening new questions which are interesting from both a mathematical and social choice point of view. In particular, we will introduce the notion of u-local optima and study it from both a theoretical and a numerically probabilistic point of view; we will also describe an algorithm that computes the universal basin of attraction of a social outcome in O(M 3log M) time (where M is the number of social outcomes).

Discriminantal arrangement, 3×3 minors of Plücker matrix and hypersurfaces in Grassmannian Gr (3, n )

We show that points in specific degree-2 hypersurfaces in the Grassmannian Gr(3,n) G r ( 3 , n ) correspond to generic arrangements of n hyperplanes in C 3 C 3 with associated discriminantal arrangement having intersections of multiplicity three in codimension two.

Blocking sets in the complement of hyperplane arrangements in projective space

Abstract It is well know that the theory of minimal blocking sets is studied by several author. Another theory which is also studied by a large number of researchers is the theory of hyperplane arrangements. We can remark that the affine space AG(n, q) is the complement of the line at infinity in PG(n, q). Then AG(n, q) can be regarded as the complement of an hyperplane arrangement in PG(n, q)! Therefore the study of blocking sets in the affine space AG(n, q) is simply the study of blocking sets in the complement of a finite arrangement in PG(n, q). In this paper the author generalizes this remark starting to study the problem of existence of blocking sets in the complement of a given hyperplane arrangement in PG(n, q). As an example she solves the problem for the case of braid arrangement. Moreover she poses significant questions on this new and interesting problem.