Singular cubic surface that is not rational

Question

It is well known that a smooth cubic surface over an algebraically closed field is rational.

And I want to know an example of a singular cubic surface that is not rational (and how to prove this). Is there any simple example?

Answer 1

Take $S:y^2z=x^3+xz^2$, considered as an affine cubic surface. It is singular at the origin, and only there. It cannot be rational, because it is a cone over the elliptic curve $E:y^2=x^3+x$ (in particular we get a dominant rational map $S \dashrightarrow E$ by mapping $(x,y,z) \mapsto (x/z,y/z)$.

If there were a rational map $\mathbb{P}^2 \dashrightarrow S$, we could compose this with the former to get a rational map $f':\mathbb{P}^2 \dashrightarrow E$. Taking a $\mathbb{P}^1 \subset \mathbb{P}^2$ containing two points with different images under $f'$, we would finally get a non-constant rational map $f:\mathbb{P}^1 \dashrightarrow E$. Since a non-constant rational map between smooth curves is a morphism, we have a contradiction, because there are no non-constant morphisms from $\mathbb{P}^1$ to an elliptic curve.