On varieties which are uniruled by lines

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