On Valuations, the Characteristic Polynomial, and Complex Subspace Arrangements

Abstract

We present a new combinatorial method to determine the characteristic polynomial of any subspace arrangement that is defined over an infinite field, generalizing the work of Blass and Sagan. Our methods stem from the theory of valuations and Groemer's integral theorem. As a corollary of our main theorem, we obtain a result of Zaslavsky about the number of chambers of a real hyperplane arrangement. The examples we consider include a family of complex subspace arrangements, which we call the divisor Dowling arrangement, whose intersection lattice generalizes that of the Dowling lattice. We also determine the characteristic polynomial of interpolations between subarrangements of the divisor Dowling arrangement, generalizing the work of Józefiak and Sagan.