Let C be a projective curve. In reading Harris’ work I often come across statements that seem to rely on the follow fact:
$H^0(C,\mathcal{O}_C(m))=H^0(\mathbb{P}^1,\mathcal{O}_{\mathbb{P}^1}(md))$, where the numbers in the parantheses refer to taking those degrees.
I have two questions:
(1) Are more restrictions on C necessary (other than just being projective)?
(2) Why is this the case?