Let $C$ a compact riemann surface of positive genus and $\omega_C$ the canonical divisor over $C$ with standard degree $2g-2$. Take on $C$ a divisor of positive degree $d$ and set $$V=H^0(C,\omega_C(D))$$ the set of section of $\omega$ with poles on $D$.
If i take an element in $V$ it has degree $2g-2+d$ but can i see an element of $V$ as a meromorphic section of $\omega_C$? And in this case must its degree be zero?
I'm confused about two both interpretation.