Let $X$ be a smooth complete complex (algebraic) 3-fold, $D$ an effective divisor on $X$, and $C$ a smooth integral curve contained in the support of $D$. Let $X'$ be the blowup of $X$ along $C$, and denote by $D'$ the strict transform of $D$, $E$ the exceptional divisor, and $C'=D' \cap E$ the (set-theoretic) intersection.


Then is it true that $C \cong C'$, $D' \cong D$, and the normal bundles $N\_{C/D} \cong N\_{C'/D'}$ under the identification of $C \cong C'$ ? Are the degrees of the normal bundles the same?



Blowing up a curve on a threefold
Let $X=\mathbb P^3$, $D$ a quadric cone and $C$ a line on $D$ that goes through the vertex of $D$. Then $D'$ is isomorphic to $D$ blown up at its vertex, in particular it is smooth, $E\simeq C\times \mathbb P^1$ and $C'$ is isomorphic to the union of one member of each of the rulings on $E$, that is $C\cup \mathbb P^1$ intersecting in a single point transversally. 


