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
$$ \gamma_{\mathcal{D}} \rightarrow \mathcal{D}, \qquad \gamma_{\mathbb{P}^2} \rightarrow \mathbb{P}^2.$$
A section of the line bundle $$ \mathcal{O}(d) = \gamma_{\mathbb{P}^2}^{* d} \rightarrow \mathbb{P}^2$$ is a homogeneous
degree $d$ polynomial. This gives us a section of the rank $3$ vector bundle
$$ \psi_1: \mathcal{D} \times \mathbb{P}^2\rightarrow
\gamma_{\mathcal{D}}^* \otimes \gamma_{\mathbb{P}^2}^{*d} \oplus
\gamma_{\mathcal{D}}^* \otimes \gamma_{\mathbb{P}^2}^{*d} \otimes T_{}^* \mathbb{P}^2 $$
given by
$$ \psi_1( [s], p) = s(p), \nabla s|_p$$
It is basically the evaluation map and the ``derivative'' at that point. We can take
$\nabla$ to be any connection. It is a fact that if $d$ is large (in fact in this
case we only need $d>1$), then $\psi_1$ is transverse to the zero set.
Note that if $s(p)=0$ then $\nabla s|_p$ is the same for every connection.
So it doesn't matter what connection we chose.
My question
is the following: Consider the following section
$$ \psi_2: \mathcal{D} \times (\mathbb{P}^2)^2 \rightarrow
\gamma_{\mathcal{D}}^* \otimes \gamma_{\mathbb{P}^2}^{*d} \oplus
\gamma_{\mathcal{D}}^* \otimes \gamma_{\mathbb{P}^2}^{*d} \otimes T_{}^* \mathbb{P}^2
\oplus \gamma_{\mathcal{D}}^* \otimes \gamma_{\mathbb{P}^2}^{*d} \oplus
\gamma_{\mathcal{D}}^* \otimes \gamma_{\mathbb{P}^2}^{*d} \otimes T_{}^* \mathbb{P}^2 $$
given by
$$ \psi_2( [s], p_1, p_2) = s(p_1), \nabla s|_{p_1}, s(p_2), \nabla s|_{p_2}$$
Is it true that the section $\psi_2$ is transverse to the zero set when
$p_1 \neq p_2$, provided $d$ is large? Is there some reference someone can point out where they
prove either this or some similar statement? My idea for proving this statement
is as follows:
Choose any two distinct points you like say $[1,0,0]$ and $[0,0,1]$ and show transversality at those points. After that, I want to argue that ``there is no loss of generality'' in assuming that those two points were $[1,0,0]$ and $[0,0,1]$, because you can always change coordinates and bring your points to these two points.
I essentially want to prove the statement for $k$ distinct points (the vector bundle there will be of rank $3k$ and the base space will be $\mathcal{D} \times (\mathbb{P}^2)^k$ (provided $d$ is large). Everything is over the complex numbers.
