$18.5.A.$ EXERCISE (ASSUMING SERRE DUALITY). Suppose $C$ is a geometrically
integral smooth projective curve over $k$.
(a) Show that $h^0(C, \omega_C)$ is the genus $g$ of $C$.
(b) Show that $\deg \omega_C = 2g – 2$. (Hint: Riemann-Roch for $\mathcal{L} = \omega_C$.)
For item b), just take item a) and apply Riemann-Roch + Serre Duality along with the hint. So:
$h^0(C, \mathcal{L})-h^0(C, \omega_C \otimes \mathcal{L}^\vee)=\text {deg} \mathcal{L}-g +1$ and do $\mathcal{L}=\omega_c \Rightarrow$
$ \Rightarrow h^0(C, \omega_C)-h^0(C, \omega_C \otimes \omega_C^{\vee})=\deg\omega_C -g +1$.
Like for a geometrically integral projective variety, $h^0(C, \mathcal{O}_C=\omega_C \otimes \omega_C^{\vee}) = 1$, then
$h^0(C, \omega_C)=\deg \omega_C -g +2$. And by item a): $\deg \omega_C=2g-2.$
How to proceed to show item a)? That's my problem.
Best Answer
From the definition of genus to a integral projective curves $C$ over an algebraically closed field, we have to:
$g=p_a(C):=h^1(C, \mathcal{O}_C)=h^0(C, \mathcal{O}_{C}^{\vee}\otimes \omega_{C})=h^0(C, \mathcal{O}_{C}\otimes \omega_{C})=h^0(C,\omega_{C}).$