A topological invariant of line arrangements

We define a new topological invariant of line arrangements in the complex projective plane. This invariant is a root of unity defined under some combinatorial restrictions for arrangements endowed with some special torsion character on the fundamental group of their complements. It is derived from the peripheral structure on the group induced by the inclusion map of the boundary of a tubular neigborhood in the exterior of the arrangement. By similarity with knot theory, it can be viewed as an analogue of linking numbers. This is an orientation-preserving invariant for ordered arrangements. We give an explicit method to compute the invariant from the equations of the arrangement, by using wiring diagrams introduced by Arvola, that encode the braid monodromy. Moreover, this invariant is a crucial ingredient to compute the depth of a character satisfying some resonant conditions, and complete the existent methods by Libgober and the first author. Finally, we compute the invariant for extended MacLane arrangements with an additional line and observe that it takes different values for the deformation classes.Comment: 19 pages, 5 figure

A class of Baker-Akhiezer arrangements

We study a class of arrangements of lines with multiplicities on the plane which admit the Chalykh–Veselov Baker–Akhiezer function. These arrangements are obtained by adding multiplicity one lines in an invariant way to any dihedral arrangement with invariant multiplicities. We describe all the Baker–Akhiezer arrangements when at most one line has multiplicity higher than 1. We study associated algebras of quasi-invariants which are isomorphic to the commutative algebras of quantum integrals for the generalized Calogero–Moser operators. We compute the Hilbert series of these algebras and we conclude that the algebras are Gorenstein. We also show that there are no other arrangements with Gorenstein algebras of quasi-invariants when at most one line has multiplicity bigger than 1

Institutions, unemployment and inactivity in the OECD countries

This paper provides new evidence on the linkages between a large array of institutional arrangements (on product, labour and financial markets) and employment performance. Our analysis includes unemployment, inactivity and jobless rates, thus allowing us to control for possible substitution effects across situations of non-employment and to check whether institutional rigidities affecting unemployment impact inactivity along the same line. To cope with common problems related to the inclusion of time-invariant institutional variables in fixed effects models, we present results of regressions based on three different estimators: PCSE, GLS and FEVD, the last one being a new procedure specifically designed to treat slowly changing variables. New institutional series are proposed, namely to account for unemployment insurance net replacement rates and employment protection legislation (EPL). Among other results, we find strong evidence of a positive effect of EPL on employment performance as well as of possible complementarities across product and labour markets regulation.unemployment ; inactivity ; institutions ; time-invariant variables

On supersolvable and nearly supersolvable line arrangements

We introduce a new class of line arrangements in the projective plane, called nearly supersolvable, and show that any arrangement in this class is either free or nearly free. More precisely, we show that the minimal degree of a Jacobian syzygy for the defining equation of the line arrangement, which is a subtle algebraic invariant, is determined in this case by the combinatorics. When such a line arrangement is nearly free, we discuss the splitting types and the jumping lines of the associated rank two vector bundle, as well as the corresponding jumping points, introduced recently by S. Marchesi and J. Vall\`es. As a by-product of our results, we get a version of the Slope Problem, looking for lower bounds on the number of slopes of the lines determined by $n$ points in the affine plane over the real or the complex numbers as well.Comment: v.3, a version of the Slope Problem, valid over the real and the complex numbers as well, is obtained, see Thm. 1.1 and Thm. 4.