Morphisms and faces of pseudo-effective cones

Abstract

Let $\pi :X\to Y$ be a morphism of projective varieties and suppose that $\alpha$ is a pseudo-effective numerical cycle class satisfying ${\pi }_{\ast }\alpha =0$. A conjecture of Debarre, Jiang, and Voisin predicts that $\alpha$ is a limit of classes of effective cycles contracted by $\pi$. We establish new cases of the conjecture for higher codimension cycles. In particular we prove a strong version when $X$ is a fourfold and $\pi$ has relative dimension one.

25 pages; revised version of half of the previous arXiv:1407.6455

References in corpus (1)

• Extremal higher codimension cycles on moduli spaces of curves