Dimension of a quasivariety

Question

Let $f_1, ...f_n$ be $n$ homogenuous polynomials, on variable $z_0, z_1, ..., z_n$. Let $V=V(f_1, ..., f_n)$ be the projective variety in $P^n$. Let $V_1=V\bigcap \{ z_0=1\}$ and $V_2=V\bigcap \{ z_0=0\}$. Then $V=V_1\bigcup V_2$. I want to get some criterion to see if $V_1$ is of dimension $>0$. For example, if $V_1$ has positive dimension, does it hold that $V_1\bigcap \{ z_i=z_j \}\neq \emptyset$ for all pairs of $i\ge1,j\ge1$?