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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!

Jesko Hüttenhain
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2 Answers2

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(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.

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  • Unfortunately, it seems like only (3) really meets my requirements, because in the other cases the bundle isn't ample. I will have a look later. – Jesko Hüttenhain Oct 29 '13 at 09:05
  • @JeskoHüttenhain: which bundle isn't ample? As I said, any line bundle of positive degree on a curve is ample, so the bundles in 1 and 2 are ample. Is that what you meant? –  Oct 29 '13 at 09:19
  • Hum, now I am confused. You wrote "Sections of K define a 2:1 cover, so K is globally generated but not ample." I think you simply mistyped then, because if that means it's a finite cover of degree $2$, then it is ample but not very ample. And that'd be what I want. In (2), you say it isn't globally generated and I thought that being globally generated is necessary for being ample. – Jesko Hüttenhain Oct 29 '13 at 09:23
  • @JeskoHüttenhain: yes, I just saw that I left out the word "very" in (1). Sorry for the confusion! For (2), no, an ample bundle need not be globally generated --- indeed as the example in (3) shows, it need not have any sections at all! –  Oct 29 '13 at 09:24
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    Fascinating. This is presenting to be exactly the learning experience I had hoped it would be. I think I will have a close look at all of those ;). – Jesko Hüttenhain Oct 29 '13 at 09:27
  • Yes, the world of ample line bundles contains many surprises! I'm glad you found the answer helpful. I would be very happy if someone could fill in the gap in (3), and if anyone else could add some more good examples to add to the list. –  Oct 29 '13 at 09:31
  • Very helpful indeed, thanks for giving all these examples. – Jesko Hüttenhain Oct 29 '13 at 09:44
  • Dear @AsalBeagDubh: as for (3), do you mean you are looking for the smallest possible very ample multiple of $K$ ? – Cantlog Nov 06 '13 at 22:22
  • @Cantlog: precisely. –  Nov 15 '13 at 11:30
  • Related: https://math.stackexchange.com/questions/95508 – Watson Feb 06 '18 at 20:33
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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$L \geq 3$.

Amit
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