I am just learning about product of projective spaces and I have some basic questions I would like to figure out. I will be working with $\mathbb{P}^m \times \mathbb{P}^m$. And by bihomogeneous form I mean $F(\mathbf{x}, \mathbf{y})$ such that $F(a\mathbf{x}, b\mathbf{y}) = a^{d_1}b^{d_2}F(\mathbf{x}, \mathbf{y})$ for some $d_1$ and $d_2 \geq 0$. (coefficients in $\mathbb{C}$)
Suppose I have a collection of bihomogeneous forms $S$.
1) What is the relation between the dimension of
affine variety
$$
\{ (\mathbf{x}, \mathbf{y}) \in \mathbb{C}^{2m+2}: F = 0 (F \in S) \}
$$
and the dimension (as a biprojective(?) variety)
$$
\{ (\mathbf{x}, \mathbf{y}) \in \mathbb{P}^{m} \times \mathbb{P}^m: F = 0 (F \in S) \}
$$? (My apologies for abusing the notation...)
2) Suppose I have a hyperplane $$ V(L) = \{ (\mathbf{x}, \mathbf{y}) \in \mathbb{P}^{m} \times \mathbb{P}^m: L(\mathbf{x}) = 0 (F \in S) \} $$ where $L$ is a non-zero linear form in the $\mathbf{x}$ variables. Does it then follow that $$ \dim (X \cap V(L)) = \dim X - 1 $$ as in the usual projective space?
Thank you very much!
Edit: I moved 1) of this question to mathovreflow https://mathoverflow.net/questions/307464/dimensions-of-a-vareity-and-its-affine-cone-in-biprojective-spaces. So now I am just asking for 2).
1) Assume $F$ is nonzero. $V(F)\subset \Bbb C^{2m+2}$ is codimension 1 (or dimension $2m+1$), so it remains to determine what $\dim V(F)\subset \Bbb P^m\times\Bbb P^m$ is. If $d_1$ or $d_2$ is zero, then $V(F)\subset \Bbb P^m\times \Bbb P^m$ may be empty, so now we assume that both are strictly positive.
Embed $\Bbb P^m\times \Bbb P^m$ into $\Bbb P^{m^2+2m}$ by the Segre embedding. WLOG, $d_1\leq d_2$. Pick $l$ a homogeneous linear polynomial in $\Bbb C[x_0,\cdots,x_m]$ so that no irreducible component of $V(F)\cap \Bbb P^m$ is contained in $V(l)$. Then $G=l^{d_2-d_1}F$ is a homogeneous polynomial on $\Bbb P^{m^2+2m}$ using the Segre coordinates so that $V(G)\cap (\Bbb P^m\times \Bbb P^m) = V(F)\subset (\Bbb P^m\times \Bbb P^m)$ on the open set $D(l)\subset \Bbb P^m\times \Bbb P^m$. Since no irreducible component of $V(F)$ is contained in $V(l)$, for each irreducible component $X_i\subset V(F)$, $\dim X_i = \dim X_i\cap D(l)$. But we may compute this final quantity in $V(G)\cap (\Bbb P^m\times \Bbb P^m)$, and since codimensions add, we have that $\dim X_i= 2m-1$.
The intersection theory of multiple such forms $F$ will follow the same pattern: whatever the dimensions of each irreducible component of $V(F)\subset \Bbb C^{2m+2}$, the dimension of the corresponding irreducible component in $\Bbb P^m\times\Bbb P^m$ will be 2 less, and then you can intersect to your heart's content.
2) No, it may be possible that $X\cap V(L)$ is empty: consider $m=1$ with $X=V(x_0)$ and $L=x_1$. In fact, you can embed fairly large varieties which do not intersect: $V(x_0,\cdots,x_k)\times\Bbb P^m$ and $V(x_{k+1},\cdots,x_m)\times\Bbb P^m$ don't intersect inside $\Bbb P^m\times\Bbb P^m$.