Manifolds with nef anticanonical bundle

Let

PARTIALLY AMPLE LINE BUNDLES ON TORIC VARIETIES

In this note we study properties of partially ample line bundles on simplicial projective toric varieties. We prove that the cone ofq-ample line bundles is a union of rational polyhedral cones, and calculate these cones in examples. We prove a restriction theorem for bigq-ample line bundles, and deduce thatq-ampleness of the anticanonical bundle is not invariant under flips. Finally we prove a Kodaira-type vanishing theorem forq-ample line bundles.

Hilbert’s 14th problem over finite fields and a conjecture on the cone of curves

We give the first examples over finite fields of rings of invariants that are not finitely generated. (The examples work over arbitrary fields, for example the rational numbers.) The group involved can be as small as three copies of the additive group. The failure of finite generation comes from certain elliptic fibrations or abelian surface fibrations having positive Mordell–Weil rank. Our work suggests a generalization of the Morrison–Kawamata cone conjecture on Calabi–Yau fiber spaces to klt Calabi–Yau pairs. We prove the conjecture in dimension two under the assumption that the anticanonical bundle is semi-ample.

ON THE CONE OF CURVES AND OF LINE BUNDLES OF A RATIONAL SURFACE

Let Z be a smooth projective rational surface. A condition that implies the polyhedrality of the cone of curves of Z is given. This one depends only on the configuration of infinitely near points associated with the morphism which provides Z from a relatively minimal model X and it holds for a wide range of surfaces whose anticanonical bundle is not ample. When the above configuration is a chain, the condition consists uniquely on deciding whether certain datum is positive. Furthermore, we study polyhedrality and regularity of the characteristic cone and of the cone of curves of Z for some particular cases.