I know very little in algebraic geometry, but I want to learn!! So I know the Riemann-Roch theorem as follow: let $$L(D)=\{\text{ meromorphic functions, s.t. }\operatorname{div}(f)\geq D \}$$ and $$O(D)=\{\text{ meromorphic differentials, s.t. }\operatorname{div}(f)\geq D\}.$$
Then $$\dim L(D^{-1})= \deg(D)-g+1+\dim(O(D)).$$ Then I would like to prove the following: if $Q$ is a quartic (4,0) on $S^2$ with one poles of most order $2$ then $Q$ is zero.
I would like to say $Q=g\,dz^4$ with $g$ meromorphic with a pole of order at most $2$. Then compute $\dim(L(D^{-1}))$ with $D=k\cdot\infty$ with $k\leq 2$ and then conclude that $\dim(L(D^{-1}))=0$ hence $g=0$.
Is the rough idea is correct? how to compute $O(D)$? is there a big difference between quartic and one form? I am ready to read any reasonable approach for somebody with a good background in differential geometry and one variable complex analysis.