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Let $X$ be a smooth projective variety over complex numbers. Let $f:Y\rightarrow Z$ be a flat family of divisors of $X$ parameterized by a smooth variety $Z$. Suppose $E$ is a reflexive sheaf on $Y$ flat over $Z$. Consider the restriction to the fibers $ E|_{f^{-1}(z)}$. Do they continue to be reflexive, or at least torsion-free? Is there a non-empty open subset of $Z$ where this happens?

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  • $\begingroup$ There was some question here recently on this subject. $\endgroup$ – Sasha Aug 17 '16 at 22:07

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