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If $X$ is a smooth projective variety and $L$ is an effective divisor that is both nef and rigid ($h^0(nD)=1$ for all $n\geq 0$), is $L$ numerically trivial?

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    $\begingroup$ It is ridiculous that this question was put on hold. $\endgroup$ – bertram Jul 11 '17 at 14:06
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    $\begingroup$ I happy with the answer so I will let the hold stand as a matter of principle. I understand the reason for context, but not every question needs a narrative explaining why the asker cares, and I decided in this situation that it would only distract from the question. $\endgroup$ – DCT Jul 13 '17 at 3:43
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No.

Let $C$ be a cubic curve in $\mathbf P^2$ and choose 9 very general points on $C$. Blowing up in these 9 points we get a smooth surface $X$ in which the proper transform $\tilde{C}$ of $C$ is an irreducible curve with $\tilde{C}^2=0$, hence it nef, but no multiple of $\tilde{C}$ moves because its normal bundle is a very general line bundle of degree 0, hence is non-torsion in the Picard group.

This is a standard example; it is written down in the paper Moving codimension-one subvarieties over finite fields by Totaro. Theorem 6.1 of that paper gives further counterexamples to the question, defined in positive characteristic.

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