Hessian ideals of a homogeneous polynomial and generalized Tjurina algebras

Abstract

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 way to determine the number of weighted homogeneous singularities of such a hypersurface.

On Hodge Theory of Singular Plane Curves

“…The direct computation by Singular also yields τ (C) = 156 and dim M (f ) 2N −3 = dim M (f ) 27 = 156. Moreover, applying Corollary 1.5 in [9], we see via a Singular computation that all singularities of the curve C are weighted homogeneous. Alternatively, there are 12 nodes, 3 in each of the 4 singular fibers of the pencils (which are triangles), and the 9 base points of the pencil, each an ordinary point of multiplicity 5.…”