This question concerns the proof for the Mumford example of an Hilbert Scheme that has a non-reduced component. I am studying the proof given on R. Hartshone, "Deformation Theory", pp. 91-94. I do not understand the following three facts.
(a) let $X\subset \mathbb{P}^{3}$ be a nonsingular cubic surface, let $L$ be the sixth exceptional curve of $X$, and $H$ the hyperplane section of $X$. Consider an irreducible curve $C$ in the linear system $|4H+2L|$. Then the following sequence is exact:
$ \qquad \qquad \qquad \qquad \qquad \quad 0\longrightarrow \mathcal{O}_{X}\longrightarrow \mathcal{O}_{X}(C) \longrightarrow \mathcal{O}_{C}(C) \longrightarrow 0 $
(b) with the same notation of (a), the following exact sequence of norml bundles is exact:
$ \qquad \qquad \qquad \qquad \qquad \qquad 0\longrightarrow \mathcal{N}_{C/X}\longrightarrow \mathcal{N}_{C} \longrightarrow \mathcal{N}_{X}|_{C} \longrightarrow 0 $
(c) with the same notation of (a), the following sequence is exact:
$ \qquad \qquad \qquad \qquad \quad 0\longrightarrow \mathcal{O}_{X}(2L-C)\longrightarrow \mathcal{O}_{X}(2L) \longrightarrow \mathcal{O}_{C}(2L) \longrightarrow 0 $
I do not know if those questions can be written in more generic terms. Bibliographical references are also welcome.
The first and last s.e.s. come from the sequence defining the structure sheaf of the curve $C$: $$0\to\mathcal I_C\to \mathcal O_X\to\mathcal O_C\to 0,$$ and we use that $\mathcal I_C = \mathcal O_X(-C)$ to get $$0\to\mathcal O_X(-C)\to \mathcal O_X\to\mathcal O_C\to 0.$$ In the first case we twist everything by $C$ (i.e., do $-\otimes\mathcal O_X(C)$) and in the last case we twist everything by $2L$.
For the middle sequence, here is a related question, with an EGA reference for good measure.