# The Stacks Project

## Tag 0AAS

Exercise 102.54.6. 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?

The code snippet corresponding to this tag is a part of the file exercises.tex and is located in lines 5671–5681 (see updates for more information).

\begin{exercise}
\label{exercise-vandermonde}
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?
\end{exercise}

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