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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!

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1 Answer 1

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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.

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