Let $S$ be a smooth complex projective surface. We consider the following two types of Hilbert schemes of $S$.
The Hilbert scheme of an ample curve $D$. Suppose that $D$ is sufficiently ample, then by Kodaira-Nakano vanishing, (one component of) the Hilbert scheme $Hilb^D(S)$ of $D$ in $S$ is a projective bundle over an abelian variety, which is thus smooth. And the Betti cohomology of $Hilb^D(S)$ is relatively easy to understand due to the above easy descriptions of $Hilb^D(S)$.
The Hilbert scheme of points. Let $n$ be a positive integer. The Hilbert scheme $Hilb^n(S)$ parametrizing length $n$ artinian subschemes of $S$ is a also smooth due to results of Fogarty. The Betti cohomology of $Hilb^n(S)$ is well-studied by many authors (Göttsche, Soergel, de Cataldo, Migliorini... Excuse me if I do not list all contributors due to my ignorance).
Now let $D_1, D_2$ be very very ample curves in $S$ and let $n$ be the intersection number of $D_1$ and $D_2$. One has the following rational map $$\phi: Hilb^{D_1}(S)\times Hilb^{D_2}(S)\dashrightarrow Hilb^n(S)$$ $$ (D_1', D_2')\mapsto D_1'\cap D_2', $$ which is defined at least for $(D_1', D_2')$ that properly intersect.
We search in the literature for an understanding of the induced map in the Betti cohomology (and the Hodge structures) $$\phi^*: H^i(Hilb^n(S),\mathbb Q)\to H^i(Hilb^{D_1}(S)\times Hilb^{D_2}(S),\mathbb Q),$$ but failed.
After trying this for some time, we still don't see possible methods to attack this very explicit problem. Any indications, references, critiques and suggestions are appreciated!
