Let $C$ be a smooth complete intersection curve in $\mathbb{P}^3_{\mathbb{C}}$, $f$ be a linear polynomial in $\mathbb{C}[X_0,...,X_3]$ which does not vanish identically on $C$. Denote by $U$ the open affine set in $\mathbb{P}^3$ defined by the non-vanishing of $f$. Is it true that for any positive integer $a$, the natural restriction morphism $H^0(C,\mathcal{O}_C(a)) \to H^0(U,\mathcal{O}_C(a)|_U)$ is surjective?
There seems to be some confusion. I want to stress on the fact that I do not take any open set and any curve. I elaborate here the motivation and some more details. As far as I understand $H^0(\mathcal{O}_C(a))$ is the degree $a$ graded part of $\mathbb{C}[X_0,...,X_3]/I_C$ when $C$ is a complete intersection curve. Assume that $f=X_0$.The question is, whether $\mathcal{O}_C(a)(U \cap C)$ is the degree $a$ graded part of $\mathbb{C}[X_1',...,X_3']/I_C$, where $X_i':=X_i/X_0$. It seems to me from the definition that this should be the case. If so, then the natural morphism sending $X_0$ to $1$ and $X_i$ to $X_i'$ should give us the surjective morphism mentioned in the original question. Is this a bit more clear? Is this wrong?
Can someone please verify this argument.