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!



diagonal cubic hypersurfaces
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.


