  In a recent paper, the first author proved the log-concavity of the
coefficients of the characteristic polynomial of a matroid realizable over a
field of characteristic 0, answering a long-standing conjecture of Read in
graph theory. We extend the proof to all realizable matroids, making progress
towards a more general conjecture of Rota-Heron-Welsh. Our proof follows from
an identification of the coefficients of the reduced characteristic polynomial
as answers to particular intersection problems on a toric variety. The
log-concavity then follows from an inequality of Hodge type.
