The Local Cohomology of the Jacobian Ring

We study the 0-th local cohomology module $H^0_{\mathbf{m}}(R(f))$ of the jacobian ring $R(f)$ of a singular reduced complex projective hypersurface $X$, by relating it to the sheaf of logarithmic vector fields along $X$. We investigate the analogies between $H^0_{\mathbf{m}}(R(f))$ 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 for $H^0_{\mathbf{m}}(R(f))$.

2010 Mathematics Subject Classification: 14B15

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