The cone of moving curves of a smooth Fano three- or fourfold

Abstract

We describe the closed cone of moving curves \({\overline{\textup{NM}}{X} \subset {N_1}(X)_{\mathbb{R}}}\) of a smooth Fano three- or fourfold X over the complex numbers \({\mathbb {C}}\) by finitely many linear equations. These equations are induced by the exceptional divisors of divisorial contractions and nef divisors on birational models of X which are obtained by flips. The proof provides an inductive way to compute the cone \({\overline{\textup{NM}}{X}}\) of moving curves and gives a description of the Mori cone of a variety X + obtained by a flip \({\phi : X \dashrightarrow X^+}\) of a small contraction on X.