This is a problem about an elliptic quartic in $\displaystyle P^{3}$

$\displaystyle Y$ is defined by

$\displaystyle x^{2} - xz - yw = 0$ and

$\displaystyle yz - xw -zw = 0$

Let $\displaystyle P$ be the point $\displaystyle (0,0,0,1)$

Let $\displaystyle f$ be the projection from $\displaystyle P$ to the plane $\displaystyle w=0$.

I need to show that $\displaystyle f$ induces an isomorphism of $\displaystyle Y - P$ with the plane cubic curve $\displaystyle K$ defined by:

$\displaystyle zy^{2} - x^{3} + xz^{2}=0$

minus the point $\displaystyle Q:=(1,0,-1)$

The point of the exercise is to prove that $\displaystyle Y$ is irreducible and nonsingular. This follows after showing that $\displaystyle f$ is an isomorphism since then we can show that $\displaystyle K$ is irreducible and nonsingular which is straightforward.

I have been trying to show that $\displaystyle f(Y)$ is contained in $\displaystyle K$ and vice versa.

Am I correct to right the $\displaystyle f(Y)$ as the equations that define $\displaystyle Y$ with $\displaystyle w==0$?

if so $\displaystyle f(Y)= x^{2} - xz = 0 = yz $ (excluding the point $\displaystyle (0,0,0)$)

But the algebra is not working out for me. Is there another approach?