Let $X\subset \mathbb{P}^3$ be a smooth cubic surface, and $L\subset X$ be a line. Let $Y\subset X$ be the subscheme cut out by the divisor $2L$.
How do we compute $H^{0}(\mathcal{N}_{Y/\mathbb{P}^3})$? (Hartshorne's Deformation Theory says it should be 9-dimensional.)
As for attempts, I've tried stuff like using \begin{align*} 0\rightarrow \mathcal{N}_{Y/X}\rightarrow \mathcal{N}_{Y/\mathbb{P}^3}\rightarrow \mathcal{N}_{X/\mathbb{P}^3}|_{Y}, \end{align*} because I know the two other terms. However, I don't know if it is short exact, and, even when I just assumed it was, I got something like $h^{0}(\mathcal{N}_{Y/\mathbb{P}^3})$ is 8 or 9, depending on whether $h^1(\mathcal{N}_{Y/\mathbb{P}^3})$ is 0 or 1 (which I don't know how to compute either).