Title: Ideals from hypersurface arrangements

Abstract:

A hyperplane arrangement in projective space is a finite set of hyperplanes. It is defined by a polynomial $f$, which is a product of linear forms defining the individual hyperplanes. It is well-known that free arrangements have favorable properties. A hyperplane arrangement is free precisely if its Jacobian ideal is Cohen-Macaulay, an algebraic property we define in the talk. The Jacobian ideal is generated by the partial derivatives of $f$.

We consider the Cohen-Macaulayness of two ideals that are related to the Jacobian ideal: its top-dimensional part and its radical. In joint work with Migliore and Schenck we showed that the related ideals are Cohen-Macaulay under a mild hypothesis. We discuss extensions for hypersurface arrangements where the polynomial $f$ is a product of irreducible forms whose degrees are at least one. These results were obtained jointly with Migliore.