This paper studies the maps of the form $Hom(X,Y)\rightarrow Hom(D,Y)$ (where $D$ is an ample divisor on $X$) and gives conditions that when it is an isomorphism. This is called non-Abelian Lefschetz hyperplane theorem. I was wondering whether one can expect something similar but dual to this be true. More precisely let $D$ be an ample divisor/complete intersection on $X$. Furthermore assume $X$ and $Y$ are projective. Then we have a closed immersion of Hom schemes of the form $\mathcal{Hom}(Y,D)\rightarrow \mathcal{Hom}(Y,X)$. What can be said about this closed immersion? Does it satisfy any form of positivity or Lefschetz type properties? (Under some suitable conditions like smoothness maybe).
Another similar question is that what can be said about the closed immersion induced on the Hilbert schemes by the immersion $D\hookrightarrow X$?
