Let $B$ parametrize a family of (reduced) curves in $\mathbb{P}^3$. Assume there exists a smooth hypersurface $X$. in $\mathbb{P}^3$ containing all the curves parametrized by $B$. In otherwords, for the universal family $\pi:\mathcal{C} \to B$ over $B$, every fiber is contained in $X$. Suppose $X$ is of degree $d$. The questions are then as follows:

1. Does there exist an open subset $U$ of $B$ such that there exists another smooth surface $X'$ in $\mathbb{P}^3$ (different from $X$) of degree $d$ containing the fibers $\pi^{-1}(t)$ for all $t \in U$? If this is not the case can we impose any criterion under which this would hold true?

2. Suppose $C$ is a curve that intersects every fiber of $\pi$ transversally. Also assume that for every $b \in B$, there exists a smooth degree $d+1$ surface in $\mathbb{P}^3$ containing the curve $\pi^{-1}(b)$ and $C$. Does there exist an open subset $U$ of $B$ such that there exists a smooth surface $Y$ in $\mathbb{P}^3$ of degree $d+1$ containing the fibers $\pi^{-1}(t)$ for all $t \in U$ and $C$?

Let $B$ parametrize a family of (reduced) curves in $\mathbb{P}^3$. Assume there exists a smooth hypersurface $X$. in $\mathbb{P}^3$ containing all the curves parametrized by $B$. In otherwords, for the universal family $\pi:\mathcal{C} \to B$ over $B$, every fiber is contained in $X$. Suppose $X$ is of degree $d$. The questions are then as follows:

1. Does there exist an open subset $U$ of $B$ such that there exists another smooth surface $X'$ in $\mathbb{P}^3$ (different from $X$) of degree $d$ containing the fibers $\pi^{-1}(t)$ for all $t \in U$? If this is not the case can we impose any criterion (for example any generacity condition) under which this would hold true?

2. Suppose $C$ is a curve that intersects every fiber of $\pi$ transversally. Also assume that for every $b \in B$, there exists a smooth degree $d+1$ surface in $\mathbb{P}^3$ containing the curve $\pi^{-1}(b)$ and $C$. Does there exist an open subset $U$ of $B$ such that there exists a smooth surface $Y$ in $\mathbb{P}^3$ of degree $d+1$ containing the fibers $\pi^{-1}(t)$ for all $t \in U$ and $C$?

Let $B$ parametrize a family of (reduced) curves in $\mathbb{P}^3$. Assume there exists a smooth hypersurface $X$. in $\mathbb{P}^3$ containing all the curves parametrized by $B$. In otherwords, for the universal family $\pi:\mathcal{C} \to B$ over $B$, every fiber is contained in $X$. Suppose $X$ is of degree $d$. The questions are then as follows:

1. Does there exist an open neighbourhood $U$ of $B$ such that there exists another smooth surface $X'$ in $\mathbb{P}^3$ (different from $X$) of degree $d$ containing the fibers $\pi^{-1}(t)$ for all $t \in U$? If this is not the case can we impose any criterion under which this would hold true?

2. Suppose $C$ is a curve that intersects every fiber of $\pi$ transversally. Also assume that for every $b \in B$, there exists a smooth degree $d+1$ surface in $\mathbb{P}^3$ containing the curve $\pi^{-1}(b)$ and $C$. Does there exist an open neighbourhood $U$ of $B$ such that there exists a smooth surface $Y$ in $\mathbb{P}^3$ of degree $d+1$ containing the fibers $\pi^{-1}(t)$ for all $t \in U$ and $C$?

Let $B$ parametrize a family of (reduced) curves in $\mathbb{P}^3$. Assume there exists a smooth hypersurface $X$. in $\mathbb{P}^3$ containing all the curves parametrized by $B$. In otherwords, for the universal family $\pi:\mathcal{C} \to B$ over $B$, every fiber is contained in $X$. Suppose $X$ is of degree $d$. The questions are then as follows:

1. Does there exist an open subset $U$ of $B$ such that there exists another smooth surface $X'$ in $\mathbb{P}^3$ (different from $X$) of degree $d$ containing the fibers $\pi^{-1}(t)$ for all $t \in U$? If this is not the case can we impose any criterion under which this would hold true?

2. Suppose $C$ is a curve that intersects every fiber of $\pi$ transversally. Also assume that for every $b \in B$, there exists a smooth degree $d+1$ surface in $\mathbb{P}^3$ containing the curve $\pi^{-1}(b)$ and $C$. Does there exist an open subset $U$ of $B$ such that there exists a smooth surface $Y$ in $\mathbb{P}^3$ of degree $d+1$ containing the fibers $\pi^{-1}(t)$ for all $t \in U$ and $C$?

Let $P$ (resp. $Q$) be the Hilbert polynomial of a curve in $\mathbb{P}^3$ (resp. a degree $d$ surface in $\mathbb{P}^3$). Denote by $Hilb_{P,Q}$ the flag Hilbert scheme corresponding to the pair of Hilbert polynomials $P,Q$. Assume that the image under the second projection map $\mathrm{pr}_2(Hilb_{P,Q})$ is at least $2$-dimensional. Under what condition can we conclude that for a general element $(C,X)$ in $Hilb_{P,Q}$, there exists an open set $U$ in the linear system $|C|$ such that the intersection of the spaces $\cap_{D \in U} I_d(D)$ is at least $2$ dimensional?

Note: For a curve $D$, $I_d(D)$ denotes the graded $d$ piece of the ideal of $D$.

Let $B$ parametrize a family of (reduced) curves in $\mathbb{P}^3$. Assume there exists a smooth hypersurface $X$. in $\mathbb{P}^3$ containing all the curves parametrized by $B$. In otherwords, for the universal family $\pi:\mathcal{C} \to B$ over $B$, every fiber is contained in $X$. Suppose $X$ is of degree $d$. The questions are then as follows:

1. Does there exist an open neighbourhood $U$ of $B$ such that there exists another smooth surface $X'$ in $\mathbb{P}^3$ (different from $X$) of degree $d$ containing the fibers $\pi^{-1}(t)$ for all $t \in U$? If this is not the case can we impose any criterion under which this would hold true?

2. Suppose $C$ is a curve that intersects every fiber of $\pi$ transversally. Also assume that for every $b \in B$, there exists a smooth degree $d+1$ surface in $\mathbb{P}^3$ containing the curve $\pi^{-1}(b)$ and $C$. Does there exist an open neighbourhood $U$ of $B$ such that there exists a smooth surface $Y$ in $\mathbb{P}^3$ of degree $d+1$ containing the fibers $\pi^{-1}(t)$ for all $t \in U$ and $C$?