Rigid and nef implies numerically trivial 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?

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.