Exercise 111.57.6. Let $k$ be a field of characteristic $p > 0$ different from $2,3$. Consider the closed subscheme $X$ of $\mathbf{P}^ n_ k$ defined by

\[ \sum \nolimits _{i = 0, \ldots , n} X_ i = 0,\quad \sum \nolimits _{i = 0, \ldots , n} X_ i^2 = 0,\quad \sum \nolimits _{i = 0, \ldots , n} X_ i^3 = 0 \]

For which pairs $(n, p)$ is this variety singular?

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