# Sections of the exceptional divisor of a blowup

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 $$\sigma = \tilde S \cap E$$?

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 $$C$$).