Is the degree homomorphism $\text{deg}: \text{Pic}(X)\to \mathbb{Z}$ surjective?

Question

Let $k$ be a field, $X$ a curve over $k$, $\operatorname{Div}(X)$ the divisor group of $X$, and $\operatorname{Pic}(X)$ the divisor class group (the Picard group) of $X$.
Consider the degree homomorphism $$ \begin{split} \deg: \operatorname{Div}(X)&\to\mathbb{Z} \\ \sum_{P\in X}n_{P}P &\mapsto \sum_{P\in X}n_{P}\cdot [k(P):k], \end{split} $$ and the induced degree homomorphism $$ \deg: \operatorname{Pic}(X)\to\mathbb{Z}. $$ Here $n_{p}\in\mathbb{Z}$, and $k(P)$ is the residue field of $P$.
Are those homomorphisms surjective?
Namely, is there necessarily exists a divisor $D\in\operatorname{Div}(X)$, with $\deg(D)=1$ ?

Answer 1

Not always. For instance, if $X$ is a conic with $X(k)=\varnothing$, the image of $\deg$ is $2\mathbb{Z}$.