Let X be a curve with genus one (i.e. elliptic) or two. Let $\omega$ be the canonical line bundle. What is the dimension of the global sections of the line bundle $\omega^{\otimes n}$?
I think this is can be done using Riemann-Roch, though I'm not sure. Thanks for any help!
Assume $g \ge 2$ for now. This can be done using Riemann-Roch, which reads as follows $$ \ell(D)-\ell(K-D) = \deg(D)+1-g. $$ Now, set $D=nK$ for $n\ge 2$. For degree reasons, $\ell(K-D)=0$, while $\deg(D) = n(2g-2)$. Rearranging gives $$\ell(nK) = n\deg(K)+1-g=2ng-2n+1-g=(2n-1)g-2n+1=(2n-1)(g-1).$$ If $g=1$, then $\omega_X$ is trivial and hence so is $\omega_X^{\otimes n}$, so the answer is $1$.