Algebra Seminar
Title and Abstract: TBD
Title and Abstract: TBD
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.
Title and Abstract: TBD
Title and Abstract: TBD
Title and Abstract: TBD
Title and Abstract: TBD
Title: Hilbert Schemes and Newton-Okounkov Bodies
Abstract: Hilbert schemes of points in the plane parametrize finite, closed subschemes of C^2 with a fixed length. In this talk, I will explain how to compute the Newton-Okounkov bodies of these Hilbert schemes, which are certain unbounded polyhedra encoding geometric information about the Hilbert schemes. Finally, I will share what is known for Hilbert schemes of points on complete toric surfaces.
Title: Lexicodes Over Principal Ideal Rings