Moving codimension-one subvarieties over finite fields

We give the first examples of nef line bundles on smooth projective varieties over finite fields which are not semi-ample. More concretely, we find smooth curves on smooth projective surfaces over finite fields such that the normal bundle has degree zero, but no positive multiple of the curve moves in a family of disjoint curves. This answers questions by Keel and Mumford. The proof uses an obstruction theory, in the spirit of homotopy theory, which links the infinitely many obstructions to moving higher and higher multiples of a given codimension-one subvariety. On 3-folds over a finite field, we find nef and big line bundles which are not semi-ample. Finally, we reprove some of the known positive results about semi-ampleness over finite fields.