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 $Hilb_{P,Q}$, the intersection of the spaces $\cap_{D \in |C|} I_d(D)$ is at least $2$ dimensional?

Note: $|C|$ denotes the linear system corresponding to $C$ in $X$ and $I_d(D)$ denotes the graded $d$ piece of $I(D)$.