For each $a\in \mathbb A^1 -\{0\}$ the scheme $X_a$ is given by $$ x=t^2-1,~ ~~y = t^3-t,~~~ z=at $$ and then we have a family over $\mathbb A^1 -\{0\}$. Our goal is to obtain the total family $\bar X$ over $\mathbb A^1$. So, we want to find the ideal $I\subset k[x,y,z,a]$ of the total family $\bar X$. A mystery to me is the way Hartshorne obtain the ideal (without any explanation) $$ I = (a^2(x+1）- z^2, ax(x+1) -yz, xz-ay, y^2,x^2(x+1)) $$
My naive attempt is simply use the third equation to eliminate $t$ and get an ideal $(a^2(x+1)-z^2, a^3y+a^2z-z^3)$. I know this is not correct but I don't know why.
Furthermore, from a scheme-theoretic point of view, the total family $\bar X$ is obtained by taking scheme-theoretic closure (see Proposition 9.8), and this can be determined by an ideal sheaf shown in Stack Project 28.6.1. I was wondering if we can explain the method here using the language of scheme.
In particular, what does a parametrization (of a curve e.g. $X_a$ above) tell us in the view of scheme? (Learning scheme-language make it hard for me to understand even a curve, and I am always trying to write down things as Spec of some rings. This is not good!)