Simple example of an ample line bundle that is not very ample

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!

Solution 1:

(Let me collect the comments into a CW answer. Others can then edit the answer to add their own favourites.)

Here are a couple of simple examples, and one non-simple one. Note that any line bundle of degree $\geq 2g+1$ on a curve of genus $g$ is very ample, so any line bundle of positive degree on a curve is ample.

1. The canonical bundle $K$ on a hyperelliptic curve of genus $\geq 2$. Sections of $K$ define a 2:1 cover, so $K$ is globally generated and ample, but not very ample. On the other hand $K^2$ is very ample: for $g \geq 3$ this is immediate by the above comment; for $g=2$ the argument is a little more involved (Hartshorne IV.3.1).

2. Any bundle $L=O_C(p)$ where $p$ is a point on an elliptic curve $C$. Riemann—Roch shows that such a bundle has a 1-dimensional space of global sections, so is not very ample, or even globally generated, but it has positive degree, so is ample. On the other hand $L^3$ is very ample, and embeds $C$ into $\mathbf{P}^2$ as a smoooth cubic, with $p$ mapping to a flex.

3. If $C$ is a curve of genus $2$, and $p,q,r$ are general points on $C$, then the bundle $L=\mathcal{O}_C(p+q-r)$$ is ample, but has no global sections at all.

4. For a trickier example, one could consider a so-called Godeaux surface. This is a particular kind of surface of general type constructed as a quotient of a quintic surface in $\mathbf{P}^3$. It has the property that the canonical bundle $K_S$ is ample, but has no global sections. For more details, see the excellent answer of Clay Cordova here. Sadly, in this case I don't know what power of $K$ is needed to obtain a very ample bundle.

Solution 2:

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$L \geq 3$.