This was a question I came up with when working on twisted cubic curve.
The twisted cubic curve in $\mathbb{A}^3$ is given by $Y=\{(t,t^2,t^3)|,t\in k\}$, we can immediately tell the defining polynomials are $y-x^2,z-x^3$ and $I(Y)=(y-x^2,z-x^3)$.
Now, it's projective closure is parametrized as $\overline{Y}=\{(s^3,s^2t,st^2,t^3)|(s,t)\in k^2-\{(0,0)\}\}$. By direct computation, I can conclude $x_0x_3-x_1x_2,x_2x_0-x_1^2,x_0x_1-x_2^2$ are equations vanishing on $\overline{Y}$ and they are linearly independent (by checking the quadratic form matrix).
Question: How can I conclude $I(\overline{Y})=(x_0x_3-x_1x_2,x_2x_0-x_1^2,x_0x_1-x_2^2)$ exactly? More generally, how can we conclude, given a parametrization of a variety, a certain set of polynomials ARE the generators of the defining ideal?
When we are having complicated parametrization (unlike affine twisted cubic curve, we can set $t=x$ and the generators of the ideal pop up immediately), we could hardly solve and show a given set of equations give certain parametrization.
I had gone through many texts mentioned the projective twisted cubic curve. All of them just mention these are the defining polynomials, but I could hardly figure out why through the question above.
In the case of projective closures, we know that the homogenization of a Groebner basis gives again a generating Groebner basis. That is, if you know the ideal in the affine case you can find the ideal of the projective closure by first finding a Groebner basis in the affine case and second homogenizing it. The result will generate the ideal of the projective closure.
Generally the problem is to show that every other polynomial satisfying the parametrization already lies in the ideal you suspect to be the ideal of your variety. There is no one method to do this. Often it involves an inductive process (by degree), ordering the monomials in a clever way, some dimension argument, change of basis,... clever tricks here and there that reduce the problem to something easier. It can be done for the twisted cubic without Groebner bases. Both methods are presented very well in section 11.10 of notes by Paul Nelson here: Algebraic Geometry notes.