# diagonal cubic hypersurfaces

At the end of

https://encyclopediaofmath.org/index.php?title=Cubic_hypersurface#References

it is stated that the diagonal cubic hypersurface

$$\sum_{i=0}^{2m+1} a_i x_i^3 = 0, m\ge 2$$

(and presumably $$a_i\not=0$$) is rational. Is this true over the complex numbers, or any field of characteristic zero? Where can I find a reference of this result? Thanks!

Yes, this is true over $$\mathbb{C}$$, and rather easy. You can assume your equation is $$\sum x_i^3=0$$. For convenience, let me call the coordinates $$x_0,\ldots ,x_m;y_0,\ldots ,y_m$$. Then your hypersurface $$X$$ contains the $$m$$-planes $$P_1: x_i=-y_i$$ and $$P_2: x_i=-\rho y_i$$, with $$\rho =e^{2\pi i/3}$$. Note that $$P_1\cap P_2=\varnothing$$. Now consider the rational map $$\varphi :P_1\times P_2 -\!--\!\!> X$$ defined as follows: given general points $$p_1\in P_1$$ and $$p_2\in P_2$$, the line $$\langle p_1,p_2\rangle$$ intersects $$X$$ in a third point $$\varphi (p_1,p_2)$$. It is easy to see that $$\varphi$$ is birational.