Let $P\_1, P\_2, Q$ denote the Hilbert scheme of a plane conic in $\mathbb{P}^3$, a quartic and a degree $d$ surface in $\mathbb{P}^3$. Then there is a natural inclusion map $i$ from Hilbert flag scheme $\mathrm{Hilb}\_{P\_1,Q}$ to $\mathrm{Hilb}\_{P\_2,Q}$ under the map, $(C,X) \mapsto (2C,X)$. 
 Then is the image under the composition of the maps, $i$ with the natural projection map $\mathrm{pr}\_2$,


$\mathrm{Hilb}\_{P\_1,Q} \to\mathrm{Hilb}\_{P\_2,Q}\to \mathrm{Hilb}\_{Q}$


an irreducible component of the image of $\mathrm{pr}\_2$? 
 If so can this result be generalized to the case when we can replace plane conic and quartic by curves $C\_1, C\_2$ such that $rC\_1$ has the same Hilbert polynomial as $C\_2$? 


Note:Hilbert flag scheme $\mathrm{Hilb}\_{P\_i,Q}$ parametrize pairs of the form $C \subset X$ where $P\_i$ is the Hilbert polynomial of $C$ and $X$ is a degree $d$ surface in $\mathbb{P}^3$.



Irreducible components of the Hilbert scheme
This question has been answered in the comments by Jason Starr. The morphism $i$ is in general not defined and the composition will in general not map to a component of the target. I am reposting this as a CW answer; if it gets upvoted, this question will not reappear on the front page.


