Hessian Ideals of a Homogeneous Polynomial and Generalized Tjurina Algebras

Using the minors in Hessian matrices, we introduce new graded algebras associated to a homogeneous polynomial. When the associated projective hypersurface has isolated singularities, these algebras are related to some new local algebras associated to isolated hypersurface singularities, which generalize their Tjurina algebras. One consequence of our results is a new very rapid way to determine the number of weighted homogeneous singularities of such a hypersurface.

2010 Mathematics Subject Classification: Primary 14J70, 13D40; Secondary 14B05, 32S05

Keywords and Phrases: projective hypersurfaces, graded algebra, Hessian matrix, weighted homogeneous singularities

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