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

2025/02/10 by Roberta Di Gennaro, Di Gennaro, Roberta, Rosa M. Miró‐Roig +1
Computer Science · Mathematics · #Algebraic Geometry (math.AG) #Computational Geometry and Mesh Generation #FOS: Mathematics #Geometric and Algebraic Topology #Mathematics and Applications

paper · pdf · doi:10.48550/arxiv.2502.06677

openalex publication_date 2025/02/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A hypersurface X⊂ \mathbb Pⁿ is said to be free if its associated sheaf T_X of vector fields tangent to X is a free \mathcal O_{\mathbb Pⁿ}-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 Pⁿ. All of them are union of hypersurfaces in a suitable pencil.

Related

• Ramblings on the freeness of affine hypersurfaces
• Saito criterion and its avatars
• Jacobian schemes arising from hypersurface arrangements in \mathbb Pⁿ
• Characterizations of freeness for equidimensional subspaces
• A generalized Saito freeness criterion