Cfrgtkky

Suppose that my initial envoirenment space is $mathbb{P}^{4}$ with homogeneous coordinates $[x_{0}:x_{1}:x_{2}:x_{3}:x_{4}]$ and I define an hypersurface with equation (for example) $h:x_{0}x_{3}+2x_{1}x_{2}-3x_{4}^2=0$.

I also define the coordinates $[y_{0}:y_{1}]$ for a second projective space $mathbb{P}^{1}$.

Now I embedd the product of the two projective varieties using the Segre embedding $mathbb{P}^{4}timesmathbb{P}^{1}hookrightarrowmathbb{P}^{9}$ with coordintes $[x_{0}y_{0},x_{1}y_{0},x_{2}y_{0},x_{3}y_{0},x_{4}y_{0},x_{0}y_{1},x_{1}y_{1},x_{2}y_{1},x_{3}y_{1},x_{4}y_{1}]$

What I understand is that now my envoirenment space is going to be $mathbb{P}^{9}$, but what I want to know is the following: which are going to be the defining equations of my hypersurface $h$ when it is embedded into $mathbb{P}^{9}$?

I have no idea about how to proceed, maybe I am misunderstanding something important. My only "reasonable" guess is that now the hypersurface could be:

$h=left{begin{matrix}
x_{0}y_{0}x_{3}y_{0}+2x_{1}y_{0}x_{2}y_{0}-3(x_{4}y_{0})^2=0\
x_{0}y_{1}x_{3}y_{1}+2x_{1}y_{1}x_{2}y_{1}-3(x_{4}y_{1})^2=0
end{matrix}right.$

(At least in this way there is kind of a "match" in the dimensions as $mathbb{P}^{3}timesmathbb{P}^{1}hookrightarrowmathbb{P}^{7}$ and generally (for a complete intersection) two equations in $mathbb{P}^{9}$ will define a $7-$manifold)