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On varieties which are uniruled by lines

A. L. Knutsen·C. Novelli·A. Sarti
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Abstract

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.

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A. L. Knutsen, C. Novelli, A. Sarti. 2006-01-12. On varieties which are uniruled by lines. https://arxiv.org/abs/math/0503161

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