Let $C$ be a smooth curve in a smooth threefold $X$. Denote by $Y$ the blowup of $X$ along $C$ with exceptional divisor $E$. Then $E \rightarrow C$ is a $\mathbb{P}^1$-bundle over $C$.
Is it true that sections of $E \rightarrow C$ correspond to smooth surfaces $S \subset X$ containing $C$?
To be more specific: If $S \subset X$ is a smooth surface containing $C$ then the strict transform of $S$ intersects $E$ along a section $\sigma$. Is the converse true? That is, given $\sigma$ a section of $E \rightarrow C$ can we always find a smooth surface $S$ such that $\sigma = \tilde S \cap E$?
Best Answer
No. For example, let $C \subset \mathbb{P}^3$ be a twisted cubic curve. Then $$ N_{C/X} \cong \mathcal{O}_C(5) \oplus \mathcal{O}_C(5), $$ and a surface $S$ smooth along $C$ corresponds to a section of the sheaf $I_C(d)$ such that the composition $$ \mathcal{O}_{\mathbb{P}^3} \to I_C(d) \to I_C/I_C^2 \otimes \mathcal{O}_{\mathbb{P}^3}(d) = N_{C/X}^\vee \otimes \mathcal{O}_C(3d) = \mathcal{O}_C(3d-5) \oplus \mathcal{O}_C(3d-5). $$ does not vanish at any point of $C$. If $$ \phi \colon N_{C/X} = \mathcal{O}_C(5) \oplus \mathcal{O}_C(5) \to \mathcal{O}_C(3d) $$ is the dual map, the corresponding section $\sigma$ is determined by $\mathrm{Ker}(\phi)$. In particular, $$ \deg(\mathrm{Ker}) = 10 - 3d \equiv 1 \bmod 3. $$ So, if you take, for instance, a section of $N_{C/X}$ that corresponds to an embedding $\mathcal{O}_C(5) \to N_{C/X}$, it does extend to a smooth surface (even to a surface smooth along $C$).