$X$ is a locally complete intersection curve with each connected component X' satisfying $H^0(X',\mathcal{O}_{X'})=k$ on $\mathbb{P}^3_k$. There is an exact sequence for every $k\geq1$
$0 \to I_X(-k) \to \mathcal{O}_{\mathbb{P}^3}(-k) \to \mathcal{O}_X(-k) \to 0$
I need to prove that $H^1(I_X(-k))=0$ for $k\geq1$, but looking at the exact sequence it is enough to show $H^0(\mathcal{O}_X (-k))=0$ since $H^0(\mathcal{O}_{\mathbb{P}^3}(-k))=H^1(\mathcal{O}_{\mathbb{P}^3}(-k))=0$.
Any suggestions on how to think about the global sections of $\mathcal{O}_X (-k)$? or on how to compute $H^1(I_X(-k))$? (at first I tried to look at the extensions)