Morphisms and faces of pseudo-effective cones

TL;DR

This paper investigates the structure of pseudo-effective cones of cycles under morphisms of projective varieties, proving new cases of a conjecture relating to limits of effective cycles contracted by the morphism.

Contribution

It establishes new cases of a conjecture on pseudo-effective classes, especially for fourfolds with morphisms of relative dimension one.

Findings

Proved the conjecture for certain higher codimension cycles.

Established the conjecture for fourfolds with relative dimension one.

Extended understanding of the behavior of pseudo-effective cones under morphisms.

Abstract

Let $\pi: X \to Y$ be a morphism of projective varieties and suppose that $\alpha$ is a pseudo-effective numerical cycle class satisfying $\pi_*\alpha = 0$. 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.