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).
$\begingroup$
$\endgroup$
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
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).
