Morphisms and faces of pseudo-effective cones

Abstract

Let π : X → Y be a morphism of projective varieties and suppose that α is a pseudo-effective numerical cycle class satisfying π * α = 0 . A conjecture of Debarre, Jiang, and Voisin predicts that α is a limit of classes of effective cycles contracted by π. We establish new cases of the conjecture for higher codimension cycles. In particular, we prove a strong version when X is a fourfold and π has relative dimension 1.