Let ${Hilb_{P,Q}}_{red}$ be the reduced scheme associated to the Hilbert flag scheme parametrizing all pairs $(C,X)$ with $C \subset X \subset \mathbb{P}^3$, where $C$ is a curve and $X$ a degree $d$ surface in $\mathbb{P}^3$. When is the natural projection map $$pr_1:{Hilb_{P,Q}}_{red} \to pr_1({Hilb_{P,Q}}_{red}) \subset {Hilb_P}_{red}$$ a flat morphism?

The fiber of the map over a curve $C$ is just $P(H^0(P^3,I_C(d)))$. So, the sufficient and necessary condition is that $\dim H^0(P^3,I_C(d))$ is constant on $Hilb_P$. 

