elliptic curve as complete intersection

Question

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.

Answer 1

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 $\mathbb{P}^n$.