If I have a finite collection of polynomials $f_i(c,z):\mathbb{C}^m\times\mathbb{C}^n\rightarrow\mathbb{C}$ where each $f_i$ is a polynomial in $z$ with some of the coefficients being elements of the vector $c$, the algebraic variety $$\mathcal{A}_c=\{z:f_i(c,z)=0\text{ for all }i\}$$ is dependent on $c$.
I conjecture that there exists an algebraic variety $\mathcal{C}\subset\mathbb{C}^m$, not equal to $\mathbb{C}^m$, and a $d\geq 0$ such that $\text{dim}(\mathcal{A}_c)=d$ if $c\not\in\mathcal{C}$ and $\text{dim}(\mathcal{A}_c)>d$ if $c\in\mathcal{C}$. $\mathcal{C}=\emptyset$ would mean $\text{dim}(\mathcal{A}_c)$ is independent of $c$ which is sensible. This is trivially true in the case of one polynomial or if $n=1$, and in the case of two polynomials quadratic in $(z_1,z_2)$ (see below).
If this or some similar result is true, could someone point me in the direction of the literature? Or any material that discuss such polynomials? I haven't found any but it seems like an elementary problem to consider, and even if this conjecture is exactly right I would like a more rich list of results for future reference. In fact I imagine the subsets $C_k\subset\mathbb{C}^m$ for which $\text{dim}(\mathcal{A}_c)=k$ form a stratification, where $C_{k+1}\subset\text{clos}(C_k)$ for all $k$, and for each $k$, $\bigcup_{l\geq k}C_l$ is a variety.
In the case where $$f_1(z_1,z_2)=az_1^2+bz_1z_2+cz_2^2=d,\\ f_2(z_1,z_2)=ez_1^2+fz_1z_2+gz_2^2=h,$$ consider them to be quadratics in $z_1$, set the roots of the quadratics to be equal, as they are given by the quadratic formula, and show that this is equivalent to $z_2$ solving a quartic whose coefficients are polynomial in $a$ through $h$. $\text{dim}(\mathcal{A}_c)>0$ if and only if this quartic is satisfied for infinitely many $z_2$ (as we need the roots to agree infinitely often), so the coefficients of the quartic must equal 0. This sketch doesn't take into account degenerate cases, $a=0$, etc., but it is only motivating.
As a side note, I would be interested to know if such a result is true if we replace $\mathbb{C}$ with $\mathbb{R}$.