Components of a variety in projective space

Question

How do we find the irreducible components of the following projective variety in $\mathbb{P}^{3}$, $V(wy-x^{2},xz-y^{2})$?

Answer 1

The intersection $\bar C$ of your two quadrics is a priori a degree 4 curve. Let's analyze it.

In the affine space $\mathbb A^3 \subset \mathbb P^3 $ corresponding to $w=1$ you are looking for the intersection of the quadrics $y-x^2=0$ and $xz-y^2$.
The relevant ideal is $I=(y-x^2,xz-y^2)=(y-x^2,x(z-x^3))$.
The affine intersection is thus $$C=V(I)=V(x,y-x^2)\cup V(y-x^2, z-x^3)=V(x,y)\cup V(y-x^2, z-x^3)=C_1\cup C_2$$ where $C_1=V(x,y)$ is a line while $C_2=V(y-x^2, z-x^3)$ is a twisted rational curve.
You must add the the intersection at infinity, that is the point(s) of $\bar C$ in the hyperplane $w=0$.
You obtain just the single point $x=y=w=0, z=1$ which is in the closure of both $C_1$ and $C_2$.

Summing up
The irreducible components of $\bar C \subset \mathbb P^3 $ are the line $\bar C_1=V^{proj }(x,y)\subset \mathbb P^3$ and the twisted cubic curve $\bar C_2=C_2\cup \lbrace [0:0:1:0]\rbrace \subset \mathbb P^3$.
Notice that $\bar C_2$ is also the image of the morphism $$\mathbb P^1 \to \mathbb P^3:[u:v]\to [uv^2:u^2v:u^3:v^3]$$

Answer 2

Using the fantastic MacAulay2 :

i1 : R = QQ[w..z]

o1 = R

o1 : PolynomialRing


i2 : primaryDecomposition ideal( w*y - x^2, x*z - y^2 )

              2                    2
o2 = {ideal (y  - x*z, x*y - w*z, x  - w*y), ideal (y, x)}