Hessian ideals of a homogeneous polynomial and generalized Tjurina algebras

TL;DR

This paper introduces new graded algebras derived from Hessian minors of homogeneous polynomials, linking them to generalized Tjurina algebras for hypersurfaces with isolated singularities, providing novel tools for analyzing singularities.

Contribution

It presents a new class of graded algebras based on Hessian minors and establishes their relation to generalized Tjurina algebras for hypersurfaces with isolated singularities.

Findings

New graded algebras associated with homogeneous polynomials

Connection between these algebras and generalized Tjurina algebras

A method to count weighted homogeneous singularities

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.