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On varieties that are uniruled by lines

Published online by Cambridge University Press:  14 July 2006

Andreas L. Knutsen
Affiliation:
Department of Mathematics, University of Oslo, PO Box 1053, Blindern, NO-0316 Oslo, Norwayandreas.knutsen@math.uio.no Dipartimento di Matematica, Università di Roma Tre, Largo San Leonardo Murialdo 1, 00146 Roma, Italy, knutsen@mat.uniroma3.it
Carla Novelli
Affiliation:
Dipartimento di Matematica, Università degli Studi di Trento, via Sommarive 14, 38050 Povo (TN), Italynovelli@science.unitn.it
Alessandra Sarti
Affiliation:
Institut für Mathematik, Johannes-Gutenberg-Universität Mainz, Staudingerweg 9, 55099 Mainz, Germanysarti@mathematik.uni-mainz.de
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Abstract

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Using the $\sharp$-minimal model program of uniruled varieties we show that, for any pair $(X, {\mathcal H})$ consisting of a reduced and irreducible variety $X$ of dimension $k \geq 3$ and a globally generated big line bundle ${\mathcal H}$ on $X$ with $d:= {\mathcal H}^k$ and $n:= h^0(X, {\mathcal H})-1$ such that $d<2(n-k)-4$, then $X$ is uniruled of ${\mathcal H}$-degree one, except if $(k,d,n)=(3,27,19)$ and a ${\sharp}$-minimal model of $(X, {\mathcal H})$ is $({\mathbb P}^3,{\mathcal O}_{{\mathbb P}^3}(3))$. We also show that the bound is optimal for threefolds.

Type
Research Article
Copyright
Foundation Compositio Mathematica 2006