On varieties that are uniruled by lines

Using the $sharp$-minimal model program of uniruled varieties we show that, for any pair $(X, {mathcal H})$ consisting of a reduced and irreducible variety $X$ of dimension $k geq 3$ and a globally generated big line bundle ${mathcal H}$ on $X$ with $d:= {mathcal H}^k$ and $n:= h^0(X, {mathcal H})-1$ such that $d<2(n-k)-4$, then $X$ is uniruled of ${mathcal H}$-degree one, except if $(k,d,n)=(3,27,19)$ and a ${sharp}$-minimal model of $(X, {mathcal H})$ is $({mathbb P}^3,{mathcal O}_{{mathbb P}^3}(3))$. We also show that the bound is optimal for threefolds.