I want to understand how to approach the solution set of $n$-dimensional system of polynomial equations in $2n$ unknowns over $\mathbb{C}$ of type
$F^k(y^1 - x^1,...,y^{n} - x^{n})=0, k=1,...,n$.
I know that in 1-dimensional case with two unknowns, the set {$(x,y)| F(y-x)=0$} is a diagonal line $y=x+a$, where $F(a)=0$.
I am aware that for dimensions two or higher this is a topic of algebraic geometry, I am just looking for directions. My question is what is the set
{$(\textbf{x},\textbf{y})|\textbf{F}(\textbf{x} - \textbf{y})=\textbf{0}$} in 2-dimensional case?