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

Roberta Di Gennaro, Rosa Maria Miró-Roig

TL;DR

The article tackles the problem of freeness for reduced hypersurfaces in projective space and expands the catalog of known free divisors by reframing Saito's criterion through Multiple Eigenschemes (ME-schemes). It develops a determinantal ME-scheme perspective that links freeness to the containment of ME-schemes inside a hypersurface and shows how freeness is preserved along pencils of hypersurfaces, enabling the construction of high-degree free divisors. Key contributions include a ME-schemes version of Saito's criterion, explicit exponent-shift formulas for free pencils, and constructive procedures to obtain large families of free hypersurfaces in $\mathbb{P}^n$ (for $n\ge 3$). The results provide a practical method to generate new free divisors beyond the planar case, with clear implications for the study of Jacobian syzygies and determinantal schemes in algebraic geometry.

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.

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

TL;DR

The article tackles the problem of freeness for reduced hypersurfaces in projective space and expands the catalog of known free divisors by reframing Saito's criterion through Multiple Eigenschemes (ME-schemes). It develops a determinantal ME-scheme perspective that links freeness to the containment of ME-schemes inside a hypersurface and shows how freeness is preserved along pencils of hypersurfaces, enabling the construction of high-degree free divisors. Key contributions include a ME-schemes version of Saito's criterion, explicit exponent-shift formulas for free pencils, and constructive procedures to obtain large families of free hypersurfaces in (for ). The results provide a practical method to generate new free divisors beyond the planar case, with clear implications for the study of Jacobian syzygies and determinantal schemes in algebraic geometry.

Abstract

A hypersurface is said to be free if its associated sheaf of vector fields tangent to is a free -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 . All of them are union of hypersurfaces in a suitable pencil.

Paper Structure

This paper contains 6 sections, 6 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The module of logarithmic derivations
  4. Eigenschemes and ME-schemes in $\mathbb{P}^n$
  5. Saito's theorem revisited
  6. Applications

Key Result

Lemma 2.5

Fix integers $d_1, \cdots ,d_{n-1}\ge 2$ and $n-1$ general partially symmetric tensors $T_i=(g_0^{i},g_1^{i},\cdots ,g_n^{i})\in (Sym^{d_i-1}{\bf k}$$^{n+1})^{\oplus n+1}$, $i=1, \cdots , n-1$. It holds:

Theorems & Definitions (19)

  • Definition 2.1
  • Example 2.2
  • Remark 2.3
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Example 2.6
  • Theorem 3.1
  • Lemma 3.2
  • proof
  • ...and 9 more