Here's a question from someone who's just found out what Serre duality (in the case of curves) is.
It occurs to me that the popular statement which can also be interpreted as the Riemann-Roch theorem corresponds to $q=1$ in the general statement as formulated here: http://en.wikipedia.org/wiki/Serre_duality. Now what about the case $q=0$, namely $\Gamma(C,\mathcal F)\cong H^1(C, \Omega_C\otimes \mathcal F^*)^*$? Why is this obvious/wrong/unimportant?