Let $f:X \to \mathbb{A}^1$ be a smooth, projective morphism of relative dimension $2$. Suppose that the fiber $X\_0:=f^{-1}(0)$ contains an irreducible rational curve, say $C$ such that the restriction of the canonical bundle $K\_{X\_0}$ of $X\_0$ to $C$ is trivial. Suppose that there exists a proper, birational morphism $g:X \to Y$ contracting $C$ i.e., $g$ induces an isomorphism (onto its image) when restricted to the complement $X \backslash C$ and $C$ maps to a point in $Y$ (one can assume that the image of $C$ is an ordinary double point in $Y$). As $C$ is rational, we know that $N\_{C|X}$ is of the form $\mathcal{O}\_C(a) \oplus \mathcal{O}\_C(b)$. What can we say about $a$ and $b$? Is there some standard technique to compute $a$ and $b$? I will be very interested to know references/literatures which study similar questions (in short normal bundles of contractible rational curves in threefolds).



Normal bundle and small contraction in threefolds
The exact sequence
$$
0 \to N\_{C/X\_0} \to N\_{C/X} \to N\_{X\_0/X}\vert\_C \to 0
$$
in this case reads as
$$
0 \to \mathcal{O}\_C(-2) \to N\_{C/X} \to \mathcal{O}\_C \to 0
$$
which means that either $(a,b) = (0,-2)$ (if the extension class is trivial) or $(a,b) = (-1,-1)$ (if the extension class is nontrivial).


