Let $E$ be the elliptic curve $x^3+y^3+z^3=0$. 


**Question.** Are there injective morphisms $E\to \mathbb CP^2$ of arbitrary high degree?


*Comments.* 1) There are injective morphisms $E\to \mathbb CP^2$ of degree $3$ (the obvious one) and of degree $6$ - whose image is the curve dual to the cubic. 2) Clearly for $\mathbb CP^1$ there are injective morphisms to $\mathbb CP^2$ of arbitrary high degree. 3) I don't know any (smooth projective) curve for which on can prove that it does not admit an injective morphism to $\mathbb CP^2$ of arbitrary high degree. 


This question is related to [Injective morphism from curves to $\mathbb CP^2$](https://mathoverflow.net/questions/122645/injective-morphism-from-curves-to-mathbb-cp2)



Injective morphism from an elliptic curve to $\mathbb CP^2$. 
The answer is yes. And more generally, one has the following result:


**Lemma:** for any elliptic curve $X$, the degree of injective morphisms $X\to \mathbb{P}^2\_{\mathbb{C}}$ (or $\mathbb{C}P^2$ if you want) is not bounded. 


**Proof:** The curve $X$ is isomorphic to a plane cubic of equation $X^3+Y^3+Z^3=\lambda XYZ$ for some $\lambda$ (Hessian form). Then, the curve has nine inflexion points. These are points $[0:1:\omega]$, $[1:0:\omega]$, $[1:\omega:0]$ with $\omega^3=1$. Choose three of them, $p\_1,p\_2,p\_3$ such that the three corresponding tangent lines $L\_1,L\_2,L\_3$ intersect at three points $L\_1\cap L\_2$, $L\_1\cap L\_3$, $L\_2\cap L\_3$, different from $p\_1,p\_2,p\_3$.


We change the coordinates and assume that $L\_1,L\_2,L\_3$ are the lines $x=0$, $y=0$ and $z=0$. The points $p\_1,p\_2,p\_3$ become points of the form $[0:1:a\_1]$, $[a\_2:0:1]$, $[1:a\_3:0]$ with $a\_1a\_2a\_3\not=0$.
Take now the birational map of $\mathbb{P}^2$ given locally by $(x,y)-->(x,x^ny)$, and globally by $[x:y:z]-->[xz^n:x^ny:z^{n+1}]$, for $n\ge 2$. Its base-points are $[0:1:0]$ and $[1:0:0]$, which do not belong to the curve. The curves contracted are $z=0$ and $x=0$, and $y=0$ is fixed. Outside of the triangle $xyz=0$, the map is an automorphism. In consequence, the image of the curve has degree $3n$ and only cuspidal singularities. This yields an injective map from $X$ to $\mathbb{P}^2$.


