I am trying to learn the properties of algebraic varieties defined by bihomogeneous polynomials. Most of the classical references (Hartshorne, Harris, etc) appears to be (at least for me) very brief on this subject. Are there any references that discuss in full details the properties of varieties $X\subset \mathbb{P}^n\times \mathbb{P}^m$ giving a number of examples?

In particular, I am confused by the following apparent paradox as far as the possibility of interpreting one bihomogeneous polynomial of bidegree $(2d,2d)$ in the variables $x_0,...,x_n;y_0,...,y_n$ as a homogeneous polynomial of degree $4d$ in the variables $z_0,...z_{2n+1}$ by means of the identification $z_0\equiv x_0,...,z_{2n+1}\equiv y_n$. To be concrete consider the variety $V_1=V(F_1)\subset \mathbb{P}^5$ defined by the following irreducible polynomial,

$$F = (x_0^{2d}+x_1^{2d}+x_2^{2d})(x_3x_4)^d - (x_3^{2d}+x_4^{2d}+x_5^{2d})(x_0x_1)^d$$,

for any integer $d\ge 1$. The dimension $d_1$ of this hypersurface $\subset \mathbb{P}^5$ is therefore $d_1 = 5-1=4$.

Alternatively, $F$ can be seen as a bihomogeneuos polynomial of bidegree $(2d,2d)$ provided we redefine $x_3\equiv y_0$, $x_4\equiv y_1$, $x_5\equiv y_2$:

$F = (x_0^{2d}+x_1^{2d}+x_2^{2d})(y_0y_1)^d - (y_0^{2d}+y_1^{2d}+y_2^{2d})(x_0x_1)^d$

Now, we have the variety $V_2=V(F)\subset \mathbb{P}^2\times\mathbb{P}^2$. By applying the Segre embedding $z_{ij}=x_i y_j$ from $\mathbb{P}^2\times\mathbb{P}^2$ to $\mathbb{P}^8$ we can conclude that the dimension $d_2$ of $V_2$ is $d_2=4-1=3$. This is because should be at least one less the dimension of the Segre embedding $\mathbb{P}^2\times\mathbb{P}^2 \rightarrow \mathbb{P}^8$ (which is four).

Question 1: Different ``embedding'' of the same polynomial can lead us to varieties with distinct dimension? This appears to be a paradox. Where I am making a mistake?

Another point is the following. It turns out that a possible solution for the polynomial $F$ is to separate the variables $x$ from $y$, i.e., we substitute the polynomial $F$ by two polynomials $G_1$ and $G_2$,

$G_1=x_0^{2d}+x_1^{2d}+x_2^{2d}-cx_0^dx_1^d,~~G_2=y_0^{2d}+y_1^{2d}+y_2^{2d}-cy_0^dy_1^d$

where $c$ is a constant. The variety $V(F)$ is now given by $V_3 = V(F) = V(G_1)\cup V(G_2)$. The dimension $d_3$ of $V_3 $ is clearly $d_3 = 1+1=2$ since we have two independent curves $\subset \mathbb{P}^2$.

Question 2: Is there any special terminology in algebraic geometry that describe subvarieties obtained in such way (separation of variables)? Any developed theory which assures us conditions for the existence of such particular solutions in bihomogeneuos varieties? Should them be called diagonal solutions or this is not the appropriate name?

Regards