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I apologize if the question is too naive or trivial:

We know that any reduced divisor in a smooth variety has Gorenstein singularities. However, I don't know if there's a cone theorem for Gorenstein varieties.

Let $|L|$ be a moving linear system on a smooth projective variety $X$. Can we find some $D \in |L|$ such that $D$ has canonical singularities? (Ideally I want something to which Cone Theorem applies). Thanks in advance.

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    $\begingroup$ If "moving" means "movable" (i.e. no fixed component), the answer is no: take the pencil $\lambda (X^3+Y^3)+\mu Z^3=0$ in $\mathbb{P}^3$. $\endgroup$
    – abx
    Commented Feb 2, 2020 at 19:49

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