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. 



Does the diffeomorphism group preserving a particular section act transitively?
I am not sure I understand the meaning of the equation $\psi\circ g=\psi$ as I don't see how to compare a section at two different points without fixing an isomorphism between the line-bundle and its pull-back under $g$. Anyway, in any possible interpretation, I don't think the group preserving $\psi$ acts transitively as it should preserve the zero locus of $\psi$. 


