Variety of negative Kodaira dimension contains a projective line Does a smooth projective variety over an algebraically closed field of negative Kodaira dimension contain a projective line?
I do not want to assume any conjectures.
 A: I remark that it is true in dimension at most $3$, over $\mathbb{C}$. (This probably isn't the easiest way to see it and is not a complete solution, but it was too short for a comment. )
In dimension $2$ it follows from the fact that all of the minimal surfaces with negative Kodaira dimension are covered by smooth rational curves (they are either $\mathbb{P}^1$-bundles or $\mathbb{P}^2$).
In dimension $3$ it follows from the existence and classification of extremal contractions for smooth $3$-folds due to Mori. Let $X$ be a smooth $3$-fold with Kodaira dimension $-\infty$ then $X$ is uniruled (Theorem 6.1.8 (FV)). Hence by a result of Miyoaka and Mori there is a non-empty Zariski open $U \subset X$ such that each point in $U$ is contained in an irreducible curve $C \subset X$ with $K_{X}.C <0$ (Theorem 6.1.4 (FV)), in particular $K_{X}$ is not nef.
(by (FV) I mean the book "Fano varieties" by Iskovskikh and Prokhorov).
Hence by the contraction theorem, there is some extremal contraction. You may consult the list of extremal contractions which can occur. In all cases; other than when the image is a point, some fibre of the extremal contraction clearly contains smooth rational curves (For conic bundles and del Pezzo bundles one needs that a general fibre is smooth, which is true). If the image is a point, $X$ is Fano $3$-fold with $b_{2}=1$. Then since Fano $3$-folds are classified we may go through each of the $17$ prime Fano $3$-folds and check there is a smooth rational curve in each.
