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Recall that a divisor $M$ on a variety $X$ is said to be semiample if $kM$ is base point free for a certain $k > 0$.

Being semiample is not a numerical property (take for example torsion and a non-torsion divisor of degree 0 on a curve, or for more sophisticated examples just look at Lazarsfeld - Positivity in Algebraic Geometry II - Ex. 10.3.3), therefore I was wondering:

It is possible to find a smooth projective variety $X$ and two effective divisors $E,D$ on $X$ such that $E \equiv D$, but $E$ is semiample while $H^0(X,kD)=\mathbb{C}$ for every $k$?

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Does Example 2.4 from answer your question?

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