Let $d$ be an integer greater than $0$. Let $P_2$ be the Hilbert polynomial of a degree $d$ surface in $\mathbb{P}^3$. Recall, the Hilbert flag scheme $\mathrm{Hilb}_{P_1,P_2}$ parametrizes curves $C$ contained in a degree $d$ surface $X$ in $\mathbb{P}^3$ such that $P_1$ is the Hilbert polynomial of $C$.

Now consider the first projection map from the above Hilbert flag scheme to the Hilbert scheme of curves with Hilbert polynomial $P_1$, denoted $\mathrm{Hilb}_{P_1}$. The question is:

For which Hilbert polynomials $P_1$ can we say that there exists at least one smooth curve in every irreducible component of the image? More simply, for which Hilbert polynomials $P_1$ can we say that there exists a smooth curve $C$ in $\mathrm{Hilb}_{P_1}$ such that it is contained in some degree $d$ surface in $\mathbb{P}^3$?