# elliptic curve as complete intersection

Let $$X$$ be an elliptic curve, which can be embedded into $$\mathbb{P}^n$$ as a curve of degree $$n+1$$. I want to show that for $$n\geq 4$$, $$X$$ can not be a complete intersection.

For hyperelliptic curves, this can be easily proved by computing the canonical sheaf. However, this does not seem to work as the degree of a canonical divisor is $$0$$. Moreover, how can we prove that $$X$$ is a set-theoretic complete intersection? Will it simply be an intersection of quadrics? Thanks in advance.

If $$C \subset \mathbb{P}^n$$ is a complete intersection of type $$(d_1,\dots,d_{n-1})$$ then $$K_C = (d_1 + \dots + d_{n-1} - n - 1)H\vert_C,$$ so if $$C$$ is elliptic, the degree of the right side is 0, hence $$(d_1 - 1) + \dots + (d_{n-1} - 1) = 2,$$ and each summand in the left side is a nonnegative integer. In particular, if $$n \ge 4$$, one of the summands must be zero, which means that the curve lies in a hyperplane, i.e., $$C \subset \mathbb{P}^{n-1}$$, not in $$\mathbb{P}^n$$.