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The question is really simple, its just terminology.

For simplicity we work on smooth algebraic surfaces and we consider the intersection form on curves on the surface.

So let $S$ be a surface and $D \in \operatorname{Pic}(S)$ a divisor. Then $D$ is said to be nef if $$ D.C \geq 0 $$ for all curves $C$ on $S$ (wasn't it Miles Reid who introduced this term?).

See http://en.wikipedia.org/wiki/Nef_line_bundle.

My question is now that with this definition, an effective curve need not be nef! This happens exactly when it has negative self-intersection, as is very well possible. But doesn't nef stand for numerically effective? That suggests that it is weaker then effective..

Please enlighten me, i guess a single line answer will do. Thanks!

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up vote 8 down vote accepted

Yes, this is confusing. Let me try to clarify.

The term "nef" was indeed coined by Reid. But it was not supposed to be an abbreviation of "numerically effective"! Rather, it was meant as an acronym standing for "numerically eventually free". (This is why sometimes, in older references, you might see it written as "NEF".)

The idea is that if $D$ is an effective Cartier divisor with the property that, for a sufficiently large natural number $m$, the linear system $|mD|$ has no basepoints, then it must be the case that $D \cdot C \geq 0$ for all curves $C$. So the nef condition is supposed to encapsulate the numerical behaviour of "eventually free" divisors.

Unfortunately, this doesn't really make things any less confusing, because a nef divisor certainly doesn't have to be eventually free (or numerically equivalent to an eventually free divisor.)

There is a quote due to Kollár somewhere, saying that nef means nef and nothing else --- in other words, one should forget where the word came from, and just learn the definition. This seems like a sensible idea to me!

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Everything you have written is correct; the terminology is slightly confusing at first, because as you said, not all effective divisors are numerically effective.

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