Morphisms and faces of pseudo-effective cones (Q2809274)

Let \(X\) be a projective variety defined over an algebraically closed field, and denote by \(\overline{\mathrm{NE}}(X)\) its Mori cone of curves. The interplay between morphisms from \(X\) and the extremal faces of \(\overline{\mathrm{NE}}(X)\) is one of the fundamental concepts of higher-dimensional geometry and becomes quite delicate when one considers higher-dimensional cycles and their pseudoeffective limits: denote by \(N_k(X)\) the vector space of \(k\)-dimensional \(\mathbb R\)-cycles modulo numerical equivalence, and by \(\overline{\mathrm{Eff}}_k(X) \subset N_k(X)\) the pseudoeffective cone, i.e. the cone obtained as the closure of the cone generated by effective cycles. Let \(\pi: X \rightarrow Y\) be a morphism, and let \(\alpha\) be a pseudoeffective class such that the push-forward \(\pi_* \alpha\) is zero. A conjecture introduced by \textit{O. Debarre} et al. [Pure Appl. Math. Q. 9, No. 4, 643--664 (2013; Zbl 1319.14010)] claims that \(\alpha\) is in the closure of the cone generated by \(k\)-dimensional subvarieties that are contracted by \(\pi\). The first main result of this paper proves this conjecture if \(\alpha\) is movable (in the sense of [the authors, ``Zariski decompositions of numerical cycle classes'', Preprint, \url{arXiv:1310.0538}] of dimension \(k \geq \dim X - \dim Y\) and \(\alpha \cdot \pi^* H^{k-(\dim X - \dim Y)+1}=0\). The second result concerns the case when \(\alpha\) is \(\pi\)-exceptional, i.e. one has \(\alpha \cdot \pi^* H^{k-(\dim X - \dim Y)}=0\). In this case the authors show that \(\alpha\) is equal to the negative part of its Zariski decomposition (in the sense of [the authors, ``Zariski decompositions of numerical cycle classes'', Preprint, \url{arXiv:1407.6455}]) and is moreover the push-forward of a pseudo-effective class on some proper subscheme of \(X\). In particular one can hope to conclude by induction on the dimension. As a special case one obtains the conjecture of Debarre, Jiang and Voisin for a morphism from a fourfold onto a threefold.