TITLE: Simple example of an ample line bundle that is not very ample
QUESTION [21 upvotes]: I am looking for a very concrete and simple example of a line bundle $L$ (on a curve or a surface) which is ample, but not very ample. I would also like that $L^{\otimes k}$ is very ample for a small $k$, in the sense that I can do a very hands-on computation and show that, say, all degree $3$ monomials in certain global sections yield an immersion into projective space. Thanks a lot in advance!

REPLY [2 votes]: Let $X$ be a nonsingular projective curve and $L$ be a line bundle on $X$. Then $L$ is ample if and only if deg$L> 0$. (see: Ample vector bundles, Hartshorne Proposition 7.1)
It is not difficult to see that a line bundle $L$ over an elliptic curve is very ample iff and only if deg$L \geq 3$.