  We associate to each toric vector bundle on a toric variety X(Delta) a
"branched cover" of the fan Delta together with a piecewise-linear function on
the branched cover. This construction generalizes the usual correspondence
between toric Cartier divisors and piecewise-linear functions. We apply this
combinatorial geometric technique to investigate the existence of resolutions
of coherent sheaves by vector bundles, using singular nonquasiprojective toric
threefolds as a testing ground. Our main new result is the construction of
complete toric threefolds that have no nontrivial toric vector bundles of rank
less than or equal to three. The combinatorial geometric sections of the paper,
which develop a theory of cone complexes and their branched covers, can be read
independently.
