Suppose I have a projective variety $X$ (for which I have explicit equations) and an involution $\iota$ on it (again, explicit). I'd like to write down explicit equations for $X/\langle \iota \rangle$, but I'm not sure how to proceed (I've seen the theoretical construction, but it didn't help me much in the task of writing down explicit equations).
For the sake of an example, say, I have a projective variety $X=V(ax^2+bxy+cy^2−z^2)⊆\mathbb{P}^2$ and I want to quotient this by the involution $ι:[x:y:z]↦[x:y:−z]$. Can someone explain to me how to get the explicit equations of $X/⟨ι⟩$? If this is not a good example, can someone provide a better example?
(An answer to the original question, but not the edited one.) A quick and dirty argument is just to observe that the quotient map identifies points if and only if they lie on the same line through $[0,0,1]$. So the quotient map is the same as the projection from that point onto $\mathbf P^1$. That is, the quotient is defined by the empty set of equations in $\mathbf P^1_{[x,y]}$.
By the way, it's a little odd to say "suppose for simplicity that this involution is fixed-point free", when it is definitely not fixed point free (over an algebraically closed field at least).