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

TL;DR

This paper revisits Saito's criterion for free hypersurfaces, reinterprets it via multiple eigenschemes, and constructs numerous new examples of free hypersurfaces in projective space, especially those forming pencils.

Contribution

It provides a new reinterpretation of Saito's criterion and introduces a method to construct large families of free hypersurfaces in projective space.

Findings

Reinterpretation of Saito's criterion using multiple eigenschemes

Construction of large families of free hypersurfaces

Examples are unions of hypersurfaces in pencils

Abstract

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.