Saito's theorem revisited and application to free pencils of hypersurfaces

A hypersurface $X\subset {\mathbb{P}}^n$ is said to be free if its associated sheaf $T_X$ of vector fields tangent to $X$ is a free ${\mathcal{O}}_{{\mathbb{P}}^n}$-module. So far few examples of free hypersurfaces are known. In this short note, we reinterpret Saito’s criterion of freeness in terms of multiple eigenschemes (ME) and as application we construct huge families of new examples of free reduced hypersurfaces in ${\mathbb{P}}^n$. All of them are union of hypersurfaces in a suitable pencil.