The Ample Cone of a Morphism.

Abstract

The main goal of this thesis is to study the geometric structure of relative ample cones for a projective morphism. On the one hand, we work out in detail some foundational results about relative cones that are difficult to find in the literature. These include fibre-wise amplitude, Nakai's and Kleiman's criteria for $R$-divisors in the relative setting. We also extend to this setting a theorem by Campana-Peternell which characterizes the boundary of the relative nef cone as being locally cut out by polynomials in a dense open subset.

On the other hand, we focus on the particular case of sequences of blowups of smooth centers of a smooth variety and analyze properties of relative numerical classes. We show that the intersection pairing $N^1(X/Y)_Ztimes N_1(X/Y)_Zlongrightarrow Z$ defines a duality of $Z$-modules. We also construct an example of a non-polyhedral relative nef cone for a sequence of blowups of smooth centers of $A^4$.