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Let $E$ be an elliptic curve. If one starts with embedding associated with invertible sheaf $\mathcal{O}(3x)$ where $x$ is some point on $E$ then one gets cubic in $\mathbb{P}^2$ and this embedding is basically given by Weierstrass function, its derivetive and constant function. My question is how one can see that invertible sheaf $\mathcal{O}(4x)$ embeds elliptic curve as intersection of two quadrics in $\mathbb{P}^3$? Are there any special functions that gives this embedding?

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This is a standard exercise and we solved it during the course by prof. Liedtke, you can find the solution here (problem. 6.2 - On another embedding of an elliptic curve).

The solution assumes some basic knowledge of algebraic geometry (ie. Riemann-Roch), but the basic idea is to compute the dimension of the space of all quadrics in $\mathbb{P}^{3}$ containing your curve $E$. One proves that there are at least two such different quadrics and then uses Bezout theorem to conclude that their intersection is exactly $E$.

If you need some more detail on any part, I'll be happy to help.

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