Abundance for uniruled pairs which are not rationally connected

Abstract

One of the central aims of the Minimal Model Program is to show that a projective log canonical pair $(X,\Delta )$ with $K_X+\Delta$ pseudoeffective has a good model, i.e. a minimal model $(Y,{\Delta }_Y)$ such that $K_Y+{\Delta }_Y$ is semiample. The goal of this paper is to show that this holds if $X$ is uniruled but not rationally connected, assuming the Minimal Model Program in dimension $\dim X-1$. Moreover, if $X$ is rationally connected, then we show that the existence of a good minimal model for $(X,\Delta )$ follows from a nonexistence conjecture for a very specific class of rationally connected pairs of Calabi–Yau type.