Big q-ample line bundles

A recent paper of Totaro developed a theory of q-ample bundles in characteristic 0. Specifically, a line bundle L on X is q-ample if for every coherent sheaf F on X, there exists an integer m ₀ such that m ≥ m ₀ implies H ⁱ (X, F ⊗ O(mL))=0 for i>q. We show that a line bundle L on a complex projective scheme X is q-ample if and only if the restriction of L to its augmented base locus is q-ample. In particular, when X is a variety and L is big but fails to be q-ample, then there exists a codimension-one subscheme D of X such that the restriction of L to D is not q-ample.