Geometry of PCF parameters in spaces of quadratic polynomials

We study algebraic relations among postcritically finite (PCF) parameters in the family $f_c(z)=z^2+c$. Ghioca, Krieger, Nguyen and Ye proved that an algebraic curve in ${\mathbb{C}}^2$ contains infinitely many PCF pairs $(c_1,c_2)$ if and only if the curve is special (i.e., the curve is a vertical or horizontal line through a PCF parameter, or the curve is the diagonal). Here we extend this result to subvarieties of ${\mathbb{C}}^n$ for any $n\ge 2$. Consequently, we obtain uniform bounds on the number of PCF pairs on non-special curves in ${\mathbb{C}}^2$ and the number of PCF parameters in real algebraic curves in $\mathbb{C}$, depending only on the degree of the curve. We also compute the optimal bound for the general curve of degree $d$.