Jacobian syzygies, stable reflexive sheaves, and Torelli properties for projective hypersurfaces with isolated singularities

TL;DR

This paper explores the connection between Jacobian syzygies, reflexive sheaves, and Torelli properties of projective hypersurfaces with isolated singularities, revealing new bounds and stability conditions.

Contribution

It establishes links between syzygies and Torelli properties, introduces a new lower bound for syzygy degrees, and analyzes stability of logarithmic sheaves for specific hypersurfaces.

Findings

Hypersurfaces with small Tjurina numbers are Torelli.

A new lower bound for syzygy degrees is provided.

Stability of reflexive sheaves is discussed for plane curves and surfaces in P^3.

Abstract

We investigate the relations between the syzygies of the Jacobian ideal of the defining equation for a projective hypersurface $V$ with isolated singularities and the Torelli properties of $V$ (in the sense of Dolgachev-Kapranov). We show in particular that hypersurfaces with a small Tjurina numbers are Torelli in this sense. When $V$ is a plane curve, or more interestingly, a surface in $P^3$, we discuss the stability of the reflexive sheaf of logarithmic vector fields along $V$. A new lower bound for the minimal degree of a syzygy associated to a 1-dimensional complete intersection is also given.