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. 



What is the simplest way to show that a section of a vector bundle is transverse to the zero set
It sounds like a multijet transversality theorem (see for example M. Golubitsky, V. Guillemin, Stable mappings and their singularities) in context of algebraic geometry. So, the answer is true --- a proof of this theorem uses only polynoms for perturbations which achieves general position.


