On varieties which are uniruled by lines

Abstract

Using the $♯$-minimal model program of uniruled varieties we show that for any pair $(X,\stackrel{˝}{)}$ consisting of a reduced and irreducible variety $X$ of dimension $k\ge 3$ and a globally generated big line bundle $\H$ on $X$ with $d:= \H^k$ and $n:=h^0(X,\stackrel{˝}{)}-1$ 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 $♯$-minimal model of $(X,\stackrel{˝}{)}$ is $(\PP^3,Ø_{\PP^3}(3))$. We also show that the bound is optimal for threefolds.

19 pages. Corrected version: lemma 2.2 in the previous version removed and substitued by lemma 2.3 in the new version