Cones of curves and of line bundles “at infinity”

We consider pencils at infinity V= F,Zd in the projective plane P2. There exists a minimal composition of point blowing ups XV→P2 eliminating the indeterminacies of the rational map P2→P1 induced by V. We systematically study the Mordell–Weil group of V, the cone of curves NE(XV)Q of XV and their relationship. In Theorem 3, we explicitly compute both objects when the projective curve defined by F is a union of curves with one place at infinity. We furthermore ask two questions on pencils in P2. Our Question 2 is answered in the affirmative in a particular case (Theorem 4), which can be viewed as a version at infinity—but for pencils of any degree—of a classical result on cubic pencils.