On varieties which are uniruled by lines

Knutsen, A. L.; Novelli, C.; Sarti, A.

On varieties which are uniruled by lines

Authors

Knutsen, A. L.

Novelli, C.

Sarti, A.

Description

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 $(\PP^3,Ø_{\PP^3}(3))$. 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