5
$\begingroup$

Let $E$ be an elliptic curve and $D_k=kp$ a divisor on $E$, where $p\in E$, for $k\in\mathbb{N}$. Then we can reconstruct $E$ from the graded ring $R(D_k)=\bigoplus_{n\geqslant0}\mathcal{L}({nD_k})$: $$E\cong \mathrm{Proj}\bigoplus_{n\geqslant0}\mathcal{L}(nkp).$$

Below is a table of what this results in for the first few values of $k$. table of veronese embeddings

(A really useful resource here is this problem sheet by Miles Reid.)

I have two questions for which I can't seem to find any reference:

  1. What happens if we carry on -- can we get an embedding for every value of $k\in\mathbb{N}$? If so, are they all different enough to be interesting, or is there a reason that most sources seem to stop after $5$?
  2. Why do some embeddings never occur in the above sequence (if this is even true)? For example, $C_7\subset\mathbb{P}(1,2,3)$.

Edit: The answer to 1. is apparently that we can keep on going, obtaining embeddings in $\mathbb{P}^{k-1}$ (since $D_k$ is very ample for $k\geqslant3$), but they do get increasingly complicated.

$\endgroup$
  • $\begingroup$ @JoeSilverman Yes, I've taken that bit out now. Thanks for spotting that! $\endgroup$ – Tim Oct 1 '15 at 23:38
3
$\begingroup$
  1. You are just considering the map defined by a linear system $|D_k|$ of degree $k$ (that $D_k=kp$ is irrelevant, any degree $k$ divisor on $E$ is linearly equivalent to $kp$ for some $p$). Any introductory book on curves will tell you that this is an embedding in $\mathbb{P}^{k-1}$ for $k\geq 3$.

  2. This gives only one embedding type for each $k$, so other types do not appear in this way.

| cite | improve this answer | |
$\endgroup$
  • $\begingroup$ (Do you mean an embedding in $\mathbb{P}^{k-1}$ for $k\geqslant3$? Also, I've been trying to find a good reference for such things, when you say any introductory book on curves, could you maybe give an example?) For the second part, why does $C_7=\mathbb{V}(x^7+y^2z+xz^2)$ not work? It's quasismooth, and some quick jottings seem to say that it's singular. $\endgroup$ – Tim Oct 2 '15 at 8:08
  • $\begingroup$ Yes, I meant $\mathbb{P}^{k-1}$, sorry -- I edited. As for a book, Hartshorne should be more than enough. For your $C_7$, we agree that it is singular, hence it cannot be the image of $E$ by an embedding. $\endgroup$ – abx Oct 2 '15 at 9:17
  • $\begingroup$ Knew that Hartshorne would cover, just didn't see it on a quick glance through -- will have a better look. My comment was actually meant to say that $C_7$ is non-singular... I can't seem to find any singular points? Though I'm sure I'm missing something obvious. $\endgroup$ – Tim Oct 2 '15 at 9:18
  • $\begingroup$ You are right, sorry again -- I re-edit. $\endgroup$ – abx Oct 2 '15 at 9:38
  • $\begingroup$ But we can show (see this question, for example) that $C_7$ has genus $1$, and we already know that it is a smooth projective curve... and thus elliptic? $\endgroup$ – Tim Oct 2 '15 at 9:42

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.