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Is there an example for $X$ a smooth projective variety, $L$ very ample line bundle, $L'$ an ample but not very ample line bundle, such that $L$ and $L'$ are numerically equivalent? (Numerically equivalent means they restrict to the same degree on each curve)

Is there an example for surfaces?

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Let me give a better answer to my previous one: Hartshorne exercise V.1.12. I think on a curve of genus >2, then the very ample divisors of degree 2g look like the canonical plus two points (you can prove this using Riemann-Roch). So even though numerical equivalence on a curve is just the degree, there is a subset of such degree divisors which are very ample. This example can be extended to a ruled surface as well probably.

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    $\begingroup$ I thought of almost the same example: concretely, let $C$ be a genus 3 curve. Let $P,Q$ be non-linearly-equivalent points on $C$. The canonical class $K$ is very ample, embedding $C$ as a quartic in $\mathbb{P}^2$. By Riemann-Roch, $K+P-Q$ has one fewer section, hence only gives a map $C \to \mathbb{P}^1$, hence clearly can't be very ample. $\endgroup$ – Jake Levinson Mar 13 '15 at 23:01

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