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

Exercise 102.54.3. Let $X = \mathbf{A}^2_\mathbf{C}$ where $\mathbf{C}$ is the field of complex numbers. A line will mean a closed subscheme of $X$ defined by one linear equation $ax + by + c = 0$ for some $a, b, c \in \mathbf{C}$ with $(a, b) \not = (0, 0)$. A curve will mean an irreducible (so nonempty) closed subscheme $C \subset X$ of dimension $1$. A quadric will mean a curve defined by one quadratic equation $ax^2 + bxy + cy^2 + dx + ey + f = 0$ for some $a, b, c, d, e, f \in \mathbf{C}$ and $(a, b, c) \not = (0, 0, 0)$.

  1. Find a curve $C$ such that every line has nonempty intersection with $C$.
  2. Find a curve $C$ such that every line and every quadric has nonempty intersection with $C$.
  3. Show that for every curve $C$ there exists another curve such that $C \cap C' = \emptyset$.

    The code snippet corresponding to this tag is a part of the file exercises.tex and is located in lines 5625–5644 (see updates for more information).

    \begin{exercise}
    \label{exercise-miss-curve}
    Let $X = \mathbf{A}^2_\mathbf{C}$ where $\mathbf{C}$ is the field
    of complex numbers. A {\it line} will mean a closed
    subscheme of $X$ defined by one linear equation $ax + by + c = 0$ for
    some $a, b, c \in \mathbf{C}$ with $(a, b) \not = (0, 0)$.
    A {\it curve} will mean an irreducible (so nonempty) closed subscheme
    $C \subset X$ of dimension $1$.
    A {\it quadric} will mean a curve defined by one
    quadratic equation $ax^2 + bxy + cy^2 + dx + ey + f = 0$
    for some $a, b, c, d, e, f \in \mathbf{C}$ and
    $(a, b, c) \not = (0, 0, 0)$.
    \begin{enumerate}
    \item Find a curve $C$ such that every line has nonempty intersection with $C$.
    \item Find a curve $C$ such that every line and every quadric has nonempty
    intersection with $C$.
    \item Show that for every curve $C$ there exists another curve
    such that $C \cap C' = \emptyset$.
    \end{enumerate}
    \end{exercise}

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