Consider over $\mathbb C$. Let $C\subseteq\mathbb P^n$ be a smooth curve. Let $g$ be the genus of $C$. Let $H$ denote the hyperplane divisor on $\mathbb P^n$. When $C$ is a canonical curve, then $K_C=H|_C$ and $n=g-1$. I wonder whether the following inverse is true.
Let $C\subseteq\mathbb P^n$ be a smooth curve. If $H|_C=K_C$, then $n=g-1$ and $C$ is a canonical curve.
In Principles of Algebraic Geometry by Griffths and Harris, bottom of p.258, it seems the above statement is used. If the above statement is wrong, can you explain why Griffths and Harris claim that $C$ is a canonical curve of genus 4?
2023-02-08
As the comment by Sasha shows, the statement is false in general. Then my question reduces to why Griffths and Harris make the claim.