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Speaker:
Juan Migliore, University of Notre Dame
Title:
The Weak Lefschetz Property and its Geometry.
Abstract:
A graded Artinian quotient $A$ of a homogeneous polynomial ring has the Weak
Lefschetz Property (WLP) if multiplication by a general linear form has maximal rank,
when viewed as a homomorphism from any component of $A$ to the next. When $A$ is
Gorenstein, the possible Hilbert functions for $A$ were completely classified by
Harima (1995) under the assumption that $A$ has WLP. Migliore and Nagel (2003)
extended this result to reduced arithmetically Gorenstein algebras (unions of linear
varieties) of any dimension whose general Artinian reduction has WLP, and they also
obtained results about maximal Betti numbers for such algebras. These results
applied to simplicial polytopes as well. Since then a great deal of interest has
focused not only on the remaining (very interesting) problems for Gorenstein
algebras, but also on the natural extension of considering algebras that are level
but not necessarily Gorenstein. These questions concern both the Artinian case and
extensions to the case of reduced algebras of higher dimension. We survey some of
these questions and results that have been obtained, including a useful geometric
criterion for the surjectivity of the multiplication by a general linear form in the
Artinian reduction of a reduced set of points.
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