Moduli of linear slices of high degree smooth hypersurfaces

We study the variation of linear sections of hypersurfaces in ${\mathbb{P}}^n$. We completely classify all plane curves, necessarily singular, whose line sections do not vary maximally in moduli. In higher dimensions, we prove that the family of hyperplane sections of any smooth degree $d$ hypersurface in ${\mathbb{P}}^n$ varies maximally for $d\ge n+3$. In the process, we generalize the classical Grauert–Mülich theorem about lines in projective space, both to $k$-planes in projective space and to free rational curves on arbitrary varieties.