Modifications of hyperplane arrangements

Abstract

This paper is concerned with five kinds of modification of hyperplane arrangements, including elementary lift, parallel translation, coning, one-element extension and restriction to a hyperplane. We show that the combinatorial classification of all hyperplane arrangements of each kind of modification will be characterized by the intersection lattice of the discriminantal or adjoint arrangement. Based on the classifications, a number of combinatorial invariants, including the unsigned coefficients of the Whitney polynomial, Whitney numbers of both kinds, face numbers and region numbers, are constants on those strata associated to the intersection lattice of the discriminantal or adjoint arrangement. Moreover, we further establish the order-preserving relations of those combinatorial invariants and a series of convolution formulae on the characteristic polynomials.