Let $\mathcal{D} \approx \mathbb{P}^{\delta_d}$, be the space of homogeneous
degree $d$ polynomials in three variables $[X,Y,Z] \in \mathbb{P}^2$ upto scaling, where
$\delta_d = \frac{d(d+3)}{2}$. Note that we have two tautological line bundles
$$ \gamma_{\mathcal{D}} \rightarrow \mathcal{D}, \qquad \gamma_{\mathbb{P}^2} \rightarrow \mathbb{P}^2.$$

Note that we have a natural section of the line bundle
$$ \psi: \mathcal{D} \times \mathbb{P}^2\rightarrow
\gamma_{\mathcal{D}}^* \otimes \gamma_{\mathbb{P}^2}^{*d} $$
given by
$$ \psi( [s], p) = s(p) $$
Let
$$ g : \mathcal{D} \times \mathbb{P}^2 \rightarrow \mathcal{D} \times \mathbb{P}^2$$
be a diffeomorphism (need not be a biholomorphism).

Consider the subgroup under which $\psi$ is invariant, i.e. $$ \psi \circ g = \psi. $$ Does this subgroup act transitively on $\mathcal{D} \times \mathbb{P}^2$ ? In other words given any two points $([s_1], p_1)$ and $([s_2], [p_2])$ does there always exist a diffeomorphism preserving $\psi$ that takes one point to the other? More generally consider the section $$ \psi_2: \mathcal{D} \times ((\mathbb{P}^2)^2 - \Delta)\rightarrow \gamma_{\mathcal{D}}^* \otimes \gamma_{\mathbb{P}^2}^{*d} \oplus \gamma_{\mathcal{D}}^* \otimes \gamma_{\mathbb{P}^2}^{*d} $$ given by $$ \psi_2 ([s], p_1, p_2) = s(p_1), s(p_2).$$ Does the subgroup of $$\mathcal{Diff} ( \mathcal{D} \times ((\mathbb{P}^2)^2 - \Delta))$$ under which $\psi_2$ is invariant act transitively on $\mathcal{D} \times ((\mathbb{P}^2)^2 - \Delta)$? Here $\Delta$ is the diagonal. Finally I have the same question for $k$ distinct points.