# Versality, bounds of global Tjurina numbers and logarithmic vector fields along hypersurfaces with isolated singularities

## TL;DR

This paper explores the relationship between syzygies, versality, and bounds on the Tjurina number of hypersurfaces with isolated singularities, leading to improved stability results for logarithmic vector fields and Torelli properties.

## Contribution

It demonstrates how bounds on the global Tjurina number enhance understanding of the stability of logarithmic vector fields and Torelli properties for hypersurfaces.

## Key findings

- Improved bounds on the stability of the sheaf of logarithmic vector fields.
- Enhanced conditions for the Torelli property of hypersurfaces.
- Deeper understanding of the relation between syzygies and hypersurface singularities.

## Abstract

We recall first the relations between the syzygies of the Jacobian ideal of the defining equation for a projective hypersurface $V$ with isolated singularities and the versality properties of $V$, as studied by du Plessis and Wall. Then we show how the bounds on the global Tjurina number of $V$ obtained by du Plessis and Wall lead to substantial improvements of our previous results on the stability of the reflexive sheaf $T\langle V\rangle$ of logarithmic vector fields along $V$, and on the Torelli property in the sense of Dolgachev-Kapranov of $V$.