Cohomological Dimension of Non-Complete Hypersurfaces.
Convex bodies associated to linear series
In his work on log-concavity of multiplicities, Okounkov showed in passing that one could associate a convex body to a linear series on a projective variety, and then use convex geometry to study such linear systems. Although Okounkov was essentially working in the classical setting of ample line bundles, it turns out that the construction goes through for an arbitrary big divisor. Moreover, this viewpoint renders transparent many basic facts about asymptotic invariants of linear series, and opens...
Curves on generic hypersurfaces
Curves on surfaces of degree 2r-... in Pr.