I have the following question on elliptic curves immersed in the projective space $\mathbb{P}^n$.

More precisey: let's take $\tau \in \mathcal{H}_1$ an element in the upper half-plane in $\mathbb{C}$ and the elliptic curve $E_{\tau}$ given by the quotient of $\mathbb{C}$ with the lattice $<1,\tau>$. Let's consider the principal polarization on $E_{\tau}$ defined by the line bundle $\mathcal{L}$ whose unique (up to constant) non-zero holomorphic section is the Riemann Theta function $\theta(z)$ defined as usual: \begin{equation} \theta(z) = \sum_{q \in \mathbb{Z}} e^{\pi i (q^2\tau + 2qz)} \end{equation}

Let's take an integer $d \geq 4$. The space of the holomorphic sections of $\mathcal{L}^{\otimes d}$ is generated, as well known, by the $d$ functions: \begin{equation} \theta_k(z) = \sum_{q \in \mathbb{Z}} e^{\pi i ((q+\frac{k}{d})^2\tau + 2(q+\frac{k}{d})z)} \end{equation} with $k \in \mathbb{Z}/d\mathbb{Z}$, and the map $\phi: E_{\tau} \longrightarrow \mathbb{P}^{d-1}$ given by this complete linear system in an embedding.

The question is the following: is true that, for the general $\tau$, every quadruple of them give an embedding of $E_{\tau}$ in $\mathbb{P}^3$ ?


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