  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.
