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.



General degree $d$ surface in $\mathbb{P}^3$
By the Noether-Lefschetz theorem, a very general surface $X\subset \mathbb{P}^3$ of degree $d\geq 4$ has $\mathrm{Pic}(X) \cong \mathbb Z$, generated by a hyperplane section $H$. But then if $C \sim aH$ and $C'\sim bH$ on $X$, we have $C\cdot C' = abd^2 > 0$, and so $C,C'$ must intersect in at least one (and generically many) point. Thus every pair of curves on $X$ intersect each other.


(This doesn't rule out that for some choices of pairs of curve classes that what you want could be true, but it does rule out that the most general possible result is true).


