I was studying the nature of singularities of cubic hypersurface on $\Bbb P^{n+1}$. The chosen form was $$f(x_0,x_1 \dots , x_{n+1})=x_0^3+x_1^3+\dotsb+x_{n+1}^3-c(x_0+x_1+\dotsb+x_{n+1} )^3=0.$$ I wonder if this equation introduces a special cubic hypersurface. Please if you know the name of it inform me.
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4$\begingroup$ When $n=2$ and $c=1$ this is called the Clebsch cubic surface. $\endgroup$– Lazzaro CampeottiCommented Apr 18 at 19:15
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