How to show that this set is a projective variety?

Question

For a specific example in an algebraic geometry project that I'm doing, I need to prove that the set $S$ of the set of elements of the form $(a^2 c:a^2d+2abc:2abd+b^2c:b^2d) $ (with $a,b,c,d \in \mathbb{C}$) that lie on $\mathbb{P}^3_{\mathbb{C}}$ is a projective variety that can be defined by the zero set of a homogeneous polynomial $f$ of degree $4$. I have two questions concerning this:

1) How does one find such a polynomial?

2) Having found the polynomial, how does one prove that $V(f)=S$?

Thank you in advance for your help!

Answer 1

Regarding 1) $y^2z^2-4xz^3-4y^3w+18xyzw-27x^2w^2=0$ is what I get from eliminating $a,b,c,d$ from $(x-a^2c,y-(a^2d+2abc),z-(2abd+b^2d),w-b^2d)$ using Macaulay2:

QQ[a,b,c,d,x,y,z,w,MonomialOrder=>Eliminate 4]
I=ideal(x-a^2*c,y-(a^2*d+2*a*b*c),z-(2*a*b*d+b^2*c),w-b^2*d)
gens gb I

As for 2) see Cox Little and O'Shea Ideals, Varieties and Algorithms Thm 1 p. 130