On varieties which are uniruled by lines

Details On varieties which are uniruled by lines On varieties which are uniruled by lines

2005-03-08

arxiv.org/abs/math/0503161v2

Mathematics

Algebraic Geometry

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

Scientific paper

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.

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