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On the Cone of Curves of an Abelian Variety

Thomas Bauer
American Journal of Mathematics
Vol. 120, No. 5 (Oct., 1998), pp. 997-1006
Stable URL: http://www.jstor.org/stable/25098632
Page Count: 10
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
On the Cone of Curves of an Abelian Variety
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Abstract

Let X be a smooth projective variety over the complex numbers and let $N_{1}(X)$ be the real vector space of 1-cycles on X modulo numerical equivalence. As usual denote by NE (X) the cone of curves on X, i.e. the convex cone in $N_{1}(X)$ generated by the effective 1-cycles. One knows by the Cone Theorem [4] that the closed cone of curves $\overline{NE}(X)$ is rational polyhedral whenever $c_{1}(X)$ is ample. For varieties X such that $c_{1}(X)$ is not ample, however, it is in general difficult to determine the structure of $\overline{NE}(X)$. The purpose of this paper is to study the cone of curves of abelian varieties. Specifically, the abelian varieties X are determined such that the closed cone $\overline{NE}(X)$ is rational polyhedral. The result can also be formulated in terms of the nef cone of X or in terms of the semi-group of effective classes in the Néron-Severi group of X.

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