I want to prove the following proposition to show the irreducibility of the twisted cubic curve. What would be the best way to prove it?
We define a homogeneous ideal $I$ in the polynomial ring ${\mathbb{C}[x,y,z,w]}$ as follows: $$I=(xz-y^2,xw-yz,yw-z^2).$$ Additionally, we define a homomorphism $\phi$ from ${\mathbb{C}[x,y,z,w]}$ to ${\mathbb{C}[s,t]}$ as follows: $$\phi(f(x,y,zw))=f(s^3,s^2t,st^2,t^3).$$ Then ${\operatorname{Ker}\phi=I}$.