Denote by $H_{P,Q}$ the flag Hilbert scheme parametrizing a pair $(C,X)$ such that $X$ is a degree $d$ surface in $\mathbb{P}^3$ with Hilbert polynomial $Q$ and $C \subset X$ is a curve with Hilbert polynomial $P$. Let $p_1$ be the natural projection map from $H_{P,Q}$ to $H_P$.

Then the questions are:

1. Is there an upper bound on the dimension of $Im(P_1)$ in terms of the degree $d$?

2. If the dimension of $H_{P,Q}$ is large (say greater than $d^2$) then if we replace $Q$ by the Hilbert polynomial of degree $d-1$ surfaces in $\mathbb{P}^3$ then is the corresponding dimension of $Im(P_1)$ is equal to the one before? This is equivalent to saying that if dimension of $H_{P,Q}$ is large then for a generic curve in $Im(P_1)$, do we have that $I_{d-1}(C) \not= \emptyset$?

3. Can we also say that the degrees of the defining equations of a generic curve in $Im(P_1)$ is the same? That is to say is there a fixed $r$-tuple of integers $(a_i)$ such that a generic curve in $Im(P_1)$ is defined by $r$ equations $Q_i$ of degree $a_i$ respectively? (of course, I do not expect $Q_i$ to be fixed)

Partial/known results and ideas of approaching this problem are most welcome.