  In this note we study properties of partially ample line bundles on
simplicial projective toric varieties. We prove that the cone of q-ample line
bundles is a union of rational polyhedral cones, and calculate these cones in
examples. We prove a restriction theorem for big q-ample line bundles, and
deduce that q-ampleness of the anticanonical bundle is not invariant under
flips. Finally we prove a Kodaira-type vanishing theorem for q-ample line
bundles.
