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Let $X$ be a smooth algebraic variety and $Y$ a reducible linear section of $X$, then is it true (or under which assumptions it is true) that $Y$ is contained in a reducible hyperplane section of $X$ (assuming that $X$ does have resucible hyperplane sections)?

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    $\begingroup$ This is already not true for quadric hypersurfaces (it is a simple exercise to write down an explicit example). $\endgroup$ – Francesco Polizzi Mar 15 at 13:03

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