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 $\mathbb{P}^{4}\times\mathbb{P}^{1}\hookrightarrow\mathbb{P}^{9}$ 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 $\mathbb{P}^{3}\times\mathbb{P}^{1}\hookrightarrow\mathbb{P}^{7}$ and generally (for a complete intersection) two equations in $\mathbb{P}^{9}$ will define a $7-$manifold)