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

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 number 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 with a 1-dimensional almost complete intersection is also given.

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

“…Their main result in this direction is [16,Theorem 5.3], which is stated as Theorem 3.1 below. We show that this result can be used to greatly strengthen two of our main results in [5], one on the stability of the reflexive sheaf T V of logarithmic vector fields along a surface V , and the other on the Torelli property in the sense of Dolgachev-Kapranov of the hypersurface V , see Theorems 3.3 and 3.6 below. Since the proofs of our results given in [5] are rather long and technical, we present here only the minor changes in these proofs, possible in view of du Plessis and Wall's result in Theorem 3.1, and leading to much stronger claims, as explained in Remarks 3.4 and 3.7.…”

“…where τ (V ), the Tjurina number of V , is the sum of the Tjurina numbers of all the singularities of V , see [5]. Jacobian syzygies and these two invariants mdr(f ) and mder(f ) occur in a number of fundamental results due to A. du Plessis and C.T.C.…”

On the syzygies and Hodge theory of nodal hypersurfaces

“…a substantially bigger number that the lower bound given in Corollary 4.3. A better lower bound for ct(D) than that given by Corollary 4.3 in the case of a nodal hypersurface with not too many nodes is given in [10]. More precisely, it is shown that…”

On stability of logarithmic tangent sheaves. Symmetric and generic determinants

“…More specifically we are interested in the following question: Does the logarithmic tangent sheaf T D determine the hypersurface D? These hypersurfaces have been called DK-Torelli in [Dim17]. Keeping this definition, we provide an extension of [Dim17, Theorem 1.5] to the case of non-isolated singularities, with a similar proof.…”

“…[FMV13,AFV16]. The stability of T D for hypersurfaces with isolated singularities was studied in [Dim17], in connection with the Torelli problem, on whether D can be reconstructed from T D . Stability of T D is a fundamental preliminary step to connect the study of equisingular deformations of D to moduli problems of sheaves over P N .…”