# Changing the Hilbert scheme of curves by adding the hyperplane section

Let $X$ be a smooth degree $d$ ($d>4$) surface in $\mathbb{P}^3$. Let $C$ be a divisor on $X$. Let $D$ be a general element of the linear series $|C+H_X|$ where $H_X$ is the hyperplane section of $X$. Denote by $P$ (resp. $P', Q$) the Hilbert polynomial of $C$ (resp. $D, X$). Assume that the first projection map from the flag Hilbert scheme $\mbox{Hilb}_{P,Q}$ to $\mbox{Hilb}_P$ is dominant onto an irreducible component containing $C$. Under what assumption, will the first projection map from $\mbox{Hilb}_{P',Q}$ to $\mbox{Hilb}_{P'}$ be dominant on an irreducible component containing $D$?