My question is: what is the precise statement of the below result?
The most precise statement I've found of the result I'm interested in is in this paper, where they say the following.
"any irreducible class in $H_2(X,\mathbb{Z})$ must have non-negative intersection with any effective class in $H_4(X,\mathbb{Z})$ unless all the representative curves are contained within the representative surfaces"
(They are considering a six-dimensional manifold, so curves sit in $H_2$ and divisors sit in $H_4$. In general, the result should concern curves and divisors in some space.)
By definition, a divisor is nef if it intersects non-negatively with all irreducible curves. The above statement is then almost saying that effective divisors are nef, except for the caveat about the curve not being allowed to be inside the divisor. Is this correct? Is this caveat the only case that prevents effective divisors from being nef?