The local cohomology of the jacobian ring

TL;DR

This paper explores the properties of the 0-th local cohomology module of the Jacobian ring of singular hypersurfaces, linking it to logarithmic vector fields and examining duality, Hodge theory, and Torelli questions.

Contribution

It introduces a novel approach connecting local cohomology of Jacobian rings with sheaves of logarithmic vector fields for singular hypersurfaces.

Findings

Establishes a relationship between local cohomology and logarithmic vector fields.

Analyzes self-duality and Hodge theoretic properties of the Jacobian ring.

Provides insights into Torelli-type questions for singular hypersurfaces.

Abstract

We study the 0-th local cohomology module of the jacobian ring of a singular reduced complex projective hypersurface X, by relating it to the sheaf of logarithmic vector field along X. We investigate the analogies between the local cohomology and the well known properties of the jacobian ring of a nonsingular hypersurface. In particular we study self-duality, Hodge theoretic and Torelli type questions.