I want to explicitly write down equations for a variety in 
$$\mathbb{P}(T\_{\mathbb{P}^2}\oplus T\_{\mathbb{P}^2})$$
What would be the natural way to do this? I know how the equations should look like over each point of $\mathbb{P}^2$ but it doesn't seem trivial to me how to "translate" them into global equations.



Projectivization of the tangent bundle
A twisted cubic on $P^3$ can be realized as the degeneration locus of a map $A\otimes O(-1) \to B\otimes O$, where $A$ and $B$ are vector spaces of dimension 2 and 3. In a relative situation you should replace vector spaces by vector bundles on the base. So, choose a pair of a vector bundles $A$ and $B$ on $P^2$ of rank 2 and three, and a map $p^\*A\otimes O(-1) \to p^\*B$, where $p:P\_{P^2}(T \oplus T) \to P^2$ is the projection and $O(-1)$ is the relative $O(-1)$ for $p$. Note that such map is given by a morphism of vector bundles
$A\otimes(T \oplus T) \to B$ on $P^2$. 


Further, a relative quadric is given by a section of a vector bundle $p^\*L\otimes O(2)$, where $L$ is a line bundle on $P^2$. Such section is given by a section of a vector bundle $L\otimes S^2T^\*$ on $P^2$. 


Altogether, you obtain the equations of your (generically) sixfold covering. 


