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    The linear system of hyperplane sections

    FE-DIVHYPSYS

    Let \(X \subseteq \PP^n\) be a projective variety over an algebraically closed field.

    1. The hyperplane sections \(X \cap H\), for hyperplanes \(H \not\supseteq X\), form a base-point-free linear system of effective Cartier divisors on \(X\).

    2. A general hyperplane section is smooth at every point where \(X\) is smooth. So it is smooth if \(X\) is smooth, and also if the singular locus of \(X\) is finite, since a general hyperplane misses finitely many points.

    Normality of \(X\) does not make a general hyperplane section smooth. The cone \(X = V(x_0 x_1 - x_2^2) \subseteq \PP^4\) is a normal threefold singular along the line \(L = V(x_0, x_1, x_2)\). Every hyperplane meets \(L\), and \(X \cap H\) is singular at the point \(L \cap H\) whenever \(H \not\supseteq L\): in the plane \(\PP^3 \cong H\) it is again a quadric cone.

    Show that the hyperplane sections of a projective variety form a base-point-free linear system of effective divisors.
    When is a general hyperplane section smooth and reduced?

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