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.

  • $\begingroup$ @JoeSilverman Yes, I've taken that bit out now. Thanks for spotting that! $\endgroup$ – Tim Oct 1 '15 at 23:38
  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.

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  • $\begingroup$ (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

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