# Example for very ample line bundle numerically equivalent to a not very ample line bundle

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?

• 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. – Jake Levinson Mar 13 '15 at 23:01