Let $X$ be a curve over a number field $k$. We have a homomorphism $cl\colon \operatorname{Emb}(X) \to \operatorname{Pic}(X)$ sending an embedding to the class of hyperplane sections of $X$ in the embedding modulo linear equivalence.
Question: If the embeddings are chosen suitably, can any divisor of $X$ be written as the difference of two hyperplane sections?
In other words, is $cl$ surjective?