# Reducible section of algebraic varieties

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)?

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