Let $F\_1(\mathbf{x}, \mathbf{y}), \ldots, F\_r(\mathbf{x}, \mathbf{y})$ be bihomogeneous polynomials with rational coefficients with bidegree $(d\_1, d\_2)$, which means 
$$
F\_i( s x\_1, \ldots, s x\_{n\_1} ; t y\_1, \ldots, t y\_{n\_2} ) = s^{d\_1} t^{d\_2} F\_i(\mathbf{x} ; \mathbf{y} ).
$$
Let $V$ be the algebraic set defined by $F\_1, \ldots, F\_r$ in $\mathbb{C}^{n\_1+ n\_2}$, and let us denote $V\_1, \ldots, V\_T$ to be the irreducible components of $V$. Then I am guessing that the following is true, but I wasn't really sure how to prove or where to find a reference for:
Let $1 \leq i \leq T$.


1. Suppose the point $(u\_1, \ldots, u\_{n\_1}, v\_{1}, \ldots, v\_{n\_2}) \in \mathbb{C}^{n\_1+n\_2}$ is contained in $V\_i$. Then for any $s,t \in \mathbb{C}$ the point $(su\_1, \ldots, su\_{n\_1}, tv\_{1}, \ldots, tv\_{n\_2})$ is also contained in $V\_i$.
2. $V\_i$ contains points all points of the form $(0,\ldots, 0, t\_1, \ldots, t\_n)$ where $t\_j \in \mathbb{C}$.


I would appreciate any comments or references. Thank you very much!



Question about properties of affine varieties defined by bihomogeneous polynomials
1. Since each $F\_i(u\_1,\dotsc,u\_{n\_1},v\_1,\dotsc,v\_{n\_2})=0$, then $F\_i(su\_1,\dotsc,su\_{n\_1},tv\_1,\dotsc,tv\_{n\_2}) = s^{d\_1}t^{d\_2} F\_i(u\_1,\dotsc,u\_{n\_1},v\_1,\dotsc,v\_{n\_2})=s^{d\_1}t^{d\_2}0 = 0$. So certainly the point $(su\_1,\dotsc,su\_{n\_1},tv\_1,\dotsc,tv\_{n\_2}) \in V$. The map $\mathbb{C}^2 \to \mathbb{C}^{n\_1+n\_2}$ given by $(s,t)\mapsto (su\_1,\dotsc,su\_{n\_1},tv\_1,\dotsc,tv\_{n\_2})$ is regular, so the image is irreducible. This irreducible set is contained in $V$, so it must be contained in an irreducible component of $V$. Since $(u\_1,\dotsc,u\_{n\_1},v\_1,\dotsc,v\_{n\_2}) \in V\_i$, so is $(su\_1,\dotsc,su\_{n\_1},tv\_1,\dotsc,tv\_{n\_2})$ for every $s,t$.
2. This is not true. It's true that every such point is in $V$ (assuming $d\_1>0$) but those points are not necessarily in each irreducible component $V\_i$ of $V$. For example, with $n\_1=n\_2=d\_1=d\_2=1$, $V=V(xy)$ has two components, one of which contains the point $(0,t)$ if and only if $t=0$.


