Abundance for uniruled pairs which are not rationally connected

TL;DR

This paper investigates the existence of good minimal models for uniruled pairs that are not rationally connected, assuming the MMP in lower dimensions, and relates rationally connected cases to a conjecture for Calabi--Yau pairs.

Contribution

It proves the existence of good models for certain uniruled pairs under MMP assumptions and links rationally connected cases to a specific nonexistence conjecture.

Findings

Good models exist for uniruled, non-rationally connected pairs assuming MMP in lower dimensions.

Rationally connected pairs' good model existence depends on a conjecture for Calabi--Yau type pairs.

Provides new insights into the structure of pairs in the Minimal Model Program.

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.