Let $H_{d_1,g_1}, H_{d_2,g_2}$ be two Hilbert schemes of curves in $\mathbb{P}^3$ with degrees $d_1, d_2$ and genus $g_1, g_2$. Denote by $H:=H_{d_1,g_1}\times H_{d_2,g_2}$ where an element in $H$ is a pair of curves $(C_1, C_2)$ with $C_i \in H_{d_i,g_i}$.

As far as I understand, a generic element of $H$ consists of pairs $(C_1,C_2)$ such that $C_1 \cap C_2 = \emptyset$.

Assume now that $g_i \not=\frac{1}{2}(d-1)(d-2)$ i.e., the Hilbert scheme do not parametrize plane curves. My question is whether we can say something similar for a degree $d$ surface in $\mathbb{P}^3$. More precisely, for a fixed $d \ge 5$, we denote by $H'$ the space parametrizing the family of all smooth degree $d$ surfaces $X$ in $\mathbb{P}^3$ such that $X$ contains at least one pair of curves $(C_1, C_2) \in H$. Then can we say that a general $X$ in $H'$ contains a non-intersection pair of curves $(C_1, C_2) \in H$. This is equivalent to saying that the dimension of an irreducible component of $H'$ is maximal (among the components of $H'$) if it parametrizes surfaces containing $(C_1, C_2) \in H$ with $C_1 \cap C_2 = \emptyset$.

Note that we need to assume $C_i$ are not plane curves because as far as I recall it gives a counter-example to the statement.