Discriminantal arrangements, fiber polytopes and formality (Q1361458)

The paper is concerned with combinatorial properties of the discriminantal arrangements introduced by \textit{Yu. I. Manin} and \textit{V. V. Shekhtman} [Group theoretical methods in physics, Vol. 1, 151-165, Utrecht, VNU Science Press (1986; Zbl 0699.58069)]. A discriminantal arrangement is associated to a given central arrangement of hyperplanes and describes its translations -- then affine arrangements -- that are not in general position. By a translation of a set of hyperplanes one understands a vector that describes the translations of each hyperplane in direction of its normal vector. Manin and Shekhtman consider only discriminantal arrangements associated to generic arrangements. The paper under review extends the definition to essential arrangements of hyperplanes. In particular, it is shown that the set of singular translations of an essential arrangement of hyperplanes is indeed again an arrangement. Also it is explained that the discriminantal arrangement to a given arrangement \(A\) is the arrangement dual to the fiber zonotope of the zonotope of \(A\). The zonotope of an arrangement is the polytope that is the Minkowsky sum of the line segments that are spanned by the unit normal vectors for each hyperplane. In the following sections the combinatorial properties of the face lattice of the fiber zonotope and the intersection lattice of a discriminantal arrangement are studied. Finally some results about formality and freeness of discriminantal arrangements are given.