I am looking for classical and elementary reason that why non-singular cubics are not rational?



non-singular cubics are not rational
As there is no precision about the dimension or about the field, I will try to give a description in some cases. There are plenty of other nice results about cubics.


1) Dimension 1: A smooth non-singular projective cubic curve is not rational. This is because the arithmetic genus, which is $g=(d-1)(d-2)/2=1$ is not equal to $0$. If you work over the complex field, you can see the points of your curve as a smooth surface definer over $\mathbb{R}$, and see that it is a torus, and not a sphere. You can also see it by computing the canonical divisor, which is trivial by adjunction.


2) Dimension 2: A smooth non-singular projective surface over an algebraic closed is rational. First you have to show that the surfaces contains at least two disjoint lines $L\_1$, $L\_2$. Then, you construct a map from $L\_1\times L\_2$ to your surface: to any general pair of points $(p\_1,p\_2)$, with $p\_i\in L\_i$ you associate the unique point $q$ in your surface such that $p\_1,p\_2,q$ are collinear.


3) Dimension 2 over a non-algebraically closed field: in general, the cubic is not rational. A simple reason could be that there is no point defined over your field. But even with a point, you can only get unirationality but not rationality. There are plenty of ways to study these cubics surfaces, for example with birational geometry (articles of Iskovkikh, Manin for example).


4) Dimension 3: a general smooth cubic threefold is not rational. This is much more complicate, but very interesting and comes from the intermediate Jacobian, see "The Intermediate Jacobian of the cubic threefold" by Clemens and Griffiths (Annals of Mathematics 1972).


5) Dimension >3: the question is open. There are plenty of rational smooth cubics but a general one should not be rational.


