explicitly constructing a certain flat family

Question

Is it possible to construct a flat family $$ \phi:\mathbb{A}_{\mathbb{C}}^8=\operatorname{Spec} \mathbb{C}[x,y,z,w,a,b,c,d]\longrightarrow \operatorname{Spec} \mathbb{C}[t_1, t_2, t_3] =\mathbb{A}_{\mathbb{C}}^3 $$ so that $$ \phi^{-1}((0,0,0)) = Z(xy+zw,ab+cw+d^2,x+a+c) $$ while $$ \phi^{-1}((t_1, t_2, t_3)) = Z(xy+zw,ab+cw,x+a+c), $$ for some $(t_1, t_2, t_3)\not=(0,0,0)$?

Answer 1

No. Suppose such a $\phi$ exists. Let $(x_0,y_0,z_0,w_0, a_0, b_0, c_0, d_0)$ be a point of $\phi^{-1}(t_1,t_2,t_3)$. Then $(x_0,y_0,z_0,w_0, a_0, b_0, c_0, 0)$ belongs to $\phi^{-1}(t_1,t_2,t_3)\cap \phi^{-1}(0,0,0)=\emptyset$!