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.