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Tag 0AAS

Chapter 102: Exercises > Section 102.54: Schemes, Final Exam, Spring 2014

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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