Let $X$ be a hyperelliptic curve with the canonical double covering map $F : X \to \mathbb{P}^1$. Let $D$ be an inverse image divisor of $F$ of degree $2$. Show that $G_p(|D|)=\{1,2\}$ if $\text{mult}_p(F)=1$ and $\{1,3\}$ if $\text{mult}_p(F)=2$.
I guess that I have to reduce the dimension gap of $L(D-np)$ and $L(D-(n-1)p)$ to the equality of the spaces $L(K-D+np)$ and $L(K-D+(n-1)p)$. But I have not been able to solve the latter.