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

Abstract

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.