Let $\mathcal{X} \to B$ be a family of smooth surfaces in $\mathbb{P}^3$. Let $\mathcal{C} \to B$ be a family of curves in $\mathbb{P}^3$. Assume that for all $b \in B$, the fiber $\mathcal{C}_b$ is contained in $\mathcal{X}_b$. It is possible that some of the fibers $\mathcal{C}_b$ is not reduced. Assume that there is a closed immersion from $\mathcal{C}$ to $\mathcal{X}$ which is also $B$-morphism. Suppose there exists a $b_0 \in B$, a closed point $x_0 \in \mathcal{C}_{b_0}$ and an open set $U_0$ of $x_0$ in $\mathcal{X}_{b_0}$ such that $\mathcal{C}_{b_0} \cap U_0$ is cut out by just one equation as a subscheme of $U_0$ i.e., it is a complete intersection in $U_0$.
Is it true that there exists an open set $U$ in $\mathcal{X}$ containing $x_0$ such that for all $b \in B$ for which $\mathcal{C}_b$ intersects $U$, $\mathcal{C}_b \cap U$ is a complete intersection curve in $\mathcal{X}_b \cap U$?
Reasonable assumption: One can assume that $B$ is smooth.
