Partially ample line bundles on toric varieties

Ample line bundles are a fundamental concept in algebraic geometry, encapsulating the central notion of positivity. A natural extension of the notion of ampleness is that of q- ampleness, for nonnegative integers q. Roughly speaking, q-ample line bundles on a variety are those which \kill cohomology in degrees above q." Line bundles of this kind have been studied by a number of authors, including Andreotti{Grauert [AG], Sommese [Som], Demailly{Peternell{Schneider [DPS], and Totaro [Tot].