Let $C$ be a curve in $\mathbb{P}^3$ which is of local complete intersection is some smooth surface in $\mathbb{P}^3$. Assume $C$ is non-reduced. What can we say about $h^0(\mathcal{O}_C)$? When can we say that $h^0(\mathcal{O}_C) \le 1$? The curve that I have is mind is of the form $2l+C'$ (seen as a divisor in a smooth surface in $\mathbb{P}^3$) where $l$ is a line and $C'$ is a reduced plane curve lying on the same plane as $l$. Any idea/reference for the direction of approaching this problem will be most appreciated.

Let $C$ be a curve in $\mathbb{P}^3$ which is of local complete intersection is some smooth surface in $\mathbb{P}^3$. Assume $C$ is non-reduced. What can we say about $h^0(\mathcal{O}_C)$? When can we say that $h^0(\mathcal{O}_C) \le 1$? The curve that I have is mind is of the form $2l+C'$ (seen as a divisor in a smooth surface in $\mathbb{P}^3$) where $l$ is a line and $C'$ is a reduced plane curve lying on the same plane as $l$. Any idea/reference for the direction of approaching this problem will be most appreciated.

EDIT Note that $h^0(\mathcal{O}_C)) \ge 1$ and is equal to $1$ if and only if $h^1(\mathcal{O}_X(-C))=h^1(\mathcal{O}_X(C)(d-4))$ vanish.