Abstract

Let X be a smooth projective variety over the complex numbers and let N1(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 N1(X) generated by the effective 1-cycles. One knows by the Cone Theorem [4] that the closed cone of curves [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /](X) is rational polyhedral whenever c1(X) is ample. For varieties X such that c1(X) is not ample, however, it is in general difficult to determine the structure of [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /](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 [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="03i" /](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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Additional Information

ISSN
1080-6377
Print ISSN
0002-9327
Pages
pp. 997-1006
Launched on MUSE
1998-10-01
Open Access
No
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