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AbstractIn this note we study properties of partially ample line bundles on simplicial projective toric varieties. We prove that the cone ofq-ample line bundles is a union of rational polyhedral cones, and calculate these cones in examples. We prove a restriction theorem for bigq-ample line bundles, and deduce thatq-ampleness of the anticanonical bundle is not invariant under flips. Finally we prove a Kodaira-type vanishing theorem forq-ample line bundles.doi:10.1017/s001708951500035x fatcat:qsgvhk3fynhqjggeravvbovq5u