  We show that the characteristic polynomial of a hyperplane arrangement can be
recovered from the class in the Grothendieck group of varieties of the
complement of the arrangement. This gives a quick proof of a theorem of Orlik
and Solomon relating the characteristic polynomial with the ranks of the
cohomology of the complement of the arrangement.
  We also show that the characteristic polynomial can be computed from the
total Chern class of the complement of the arrangement; this has also been
observed by Huh. In the case of free arrangements, we prove that this Chern
class agrees with the Chern class of the dual of a bundle of differential forms
with logarithmic poles along the hyperplanes in the arrangement; this follows
from work of Mustata and Schenck. We conjecture that this relation holds for
all free divisors.
  We give an explicit relation between the characteristic polynomial of an
arrangement and the Segre class of its singularity (`Jacobian') subscheme. This
gives a variant of a recent result of Wakefield and Yoshinaga, and shows that
the Segre class of the singularity subscheme of an arrangement together with
the degree of the arrangement determine the ranks of the cohomology of its
complement.
  We also discuss the positivity of the Chern classes of hyperplane
arrangements: we give a combinatorial interpretation of this phenomenon, and we
discuss the cases of generic and free arrangements.
