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We investigate the relations between the syzygies of the Jacobian ideal ofthe defining equation for a projective hypersurface $V$ with isolatedsingularities and the Torelli properties of $V$ (in the sense ofDolgachev-Kapranov). We show in particular that hypersurfaces with a smallTjurina numbers are Torelli in this sense. When $V$ is a plane curve, or moreinterestingly, a surface in $P^3$, we discuss the stability of the reflexivesheaf of logarithmic vector fields along $V$. A new lower bound for the minimaldegree of a syzygy associated to a 1-dimensional complete intersection is alsogiven.

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