## Access

You are not currently logged in.

Access your personal account or get JSTOR access through your library or other institution:

## If You Use a Screen Reader

This content is available through Read Online (Free) program, which relies on page scans. 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

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
Preview not available

## 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.

• 997
• 998
• 999
• 1000
• 1001
• 1002
• 1003
• 1004
• 1005
• 1006