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Sorry if this question is not appropriate for this site, but I haven't got an answer on stackexchange. It's well known that there are divisors (on a normal projective variety over the complex numbers) that are big (is a sum of ample and effective), but not ample. However, if we change ample to very ample, does this hold true?

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Take a smooth projective surface with a curve $D$ with $D^2<0$. If $H$ is very ample, for all large $n$, $H+nD$ is not ample.

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