Suppose $X$ is a curve over $k$ (say an algebraically closed field) and we know the genus of $X$ is $g$. Let $K$ be a canonical divisor on $X$. Is there anything we can say about $H^0(X,dK)$ for positive integers $d$? Of course by definition, we have that $H^0(X,K)=g$, but do we know anything else? If for example, $g=0$, then we can use Riemann Roch to get that $h^0(dK)=\deg (dK) +1$, but I'm curious to know if we can say anything more specific.
Edit: Via Riemann Roch, $h^0(dK)=h^0(K-dK)+\deg(dK)+1-g=h^0(K-dK)+d(2g-2)+1-g$, but I'm not sure how to get an explicit description with this.
Further edit: By Remark 1.3.2 in Ch.IV.1 of Hartshorne, we have that $h^0(dK)=d\deg K-g+1=d(2g-2)-g+1$ as soon as $d\deg K > \deg K$, that is, as soon as $d(2g-2)>2g-2$. For genus $0$, this is never satisfied, so I don't know how to get a description.