Ample subschemes and partially positive line bundles

Abstract

A common theme in algebraic geometry is that geometric properties of an algebraic variety are reflected in the subvarieties that are 'positively embedded' in it. Here the case of subvarieties of codimension one is classical and also fundamental to algebraic geometry: divisors correspond to line bundles and ample line bundles classify projective embeddings. However, subvarieties and algebraic cycles of higher codimension are regarded as much more complicated objects and not well understood in general. The first paper of the thesis introduces a notion of ampleness for subschemes of arbitrary codimension, generalising the notion of an ample divisor. In short, a subscheme is defined to be ample if the exceptional divisor on the blow-up along the subscheme satisfies a certain partial positivity condition, namely that its asymptotic cohomology groups vanish in certain degrees. It is shown that such subschemes share several geometric properties with complete intersections of ample divisors. For example, the Lefschetz hyperplane theorem on rational cohomology holds and the cycle class of an ample subvariety is numerically positive. Using these results, we construct counterexamples to a conjecture of Demailly- Peternell- Schneider on the converse of the Andreotti-Grauert vanishing theorem in complex geometry. The second paper studies the birational structure of hypersurfaces in products of projective spaces. These hypersurfaces are in many respects simple varieties, yet they provide many interesting examples of birational geometry phenomena. For example, they may have infinite birational automorphism groups. In the case of hypersurfaces in JP>m x JP>n, we study their nef, movable and effective cones and determine when they are Mori dream spaces.