Why do we call the cone of curves(effective one cycles) on a variety $X$ as $NE(X)$, what does $NE$ stand for?

  • 3
    $\begingroup$ I googled cone of curves NE(X) "notation" and found: Numerically effective (Considering the effort expended, I don't think this is worth posting as an answer...) $\endgroup$ Oct 13 '15 at 3:42
  • $\begingroup$ @BenjaminDickman And I am also wondering why effective divisor is sometimes denote by $N$, for example in that positivity book, big divisor is characterized as divisors num equiv to $A+N$, $A$ stands for ample and $N$ stands for effective divisor. $\endgroup$
    – Qixiao
    Oct 13 '15 at 4:32
  • $\begingroup$ SInce the question appears to be getting helpful answers, I can't resist mentioning F U N E X ? (for those fans of a certain vintage of British comedy). On a more serious note, it does seem that the stated question is easily answered by searching online $\endgroup$
    – Yemon Choi
    Oct 13 '15 at 14:16
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    $\begingroup$ I can't resist mentioning that a small band of rebels has been trying to replace $NE(X)$ with the notation $\operatorname{Curv}(X)$, which has the advantage that actually gives some clue to what it denotes. See for example: math.ucla.edu/~totaro/papers/public_html/cone.pdf As far as I can tell, it's not making much difference. $\endgroup$ Oct 14 '15 at 8:52
  • $\begingroup$ There is also the notation $\operatorname{Eff}_1(X)$ ('effective 1-cycles'), which better generalizes to cycles of higher dimensions. $\endgroup$ Oct 21 '15 at 22:44

As far as I know, the notation appears first in Mori's landmark paper of 1982. He uses $N(X)$ for the group of 1-cycles modulo Numerical equivalence (tensored with $\mathbb{R}$), and then, quite naturally, $NE(X)$ for the convex cone spanned by classes of Effective cycles in $N(X)$.


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