A tour of the Weak and Strong Lefschetz Properties

Abstract

An artinian graded algebra, $A$, is said to have the Weak Lefschetz property (WLP) if multiplication by a general linear form has maximal rank in every degree. A vast quantity of work has been done studying and applying this property, touching on numerous and diverse areas of algebraic geometry, commutative algebra, and combinatorics. Amazingly, though, much of this work has a "common ancestor" in a theorem originally due to Stanley, although subsequently reproved by others. In this expository paper we describe the different directions in which research has moved starting with this theorem, and we discuss some of the open questions that continue to motivate current research.

18 pages, 3 figures. Added three references in section 6 and in the bibliography

References in corpus (9)

• Inverse systems, Gelfand-Tsetlin patterns and the weak Lefschetz property
• The Lefschetz properties of monomial complete intersections in positive characteristic
• Syzygy Bundles on P^2 and the Weak Lefschetz Property
• The strong Lefschetz property and simple extensions
• Hyperplane sections and the subtlety of the Lefschetz properties
• On the shape of a pure O-sequence
• The weak Lefschetz property for Artinian graded rings and basic sequences
• On Unimodality of Hilbert Functions of Gorenstein Artin Algebras of Embedding Dimension Four
• Gorenstein algebras presented by quadrics

Cited by in corpus (10)

• The Lefschetz properties of monomial complete intersections in positive characteristic
• On the structure of monomial complete intersections in positive characteristic
• On decomposing monomial algebras with the Lefschetz properties
• Enumerations of lozenge tilings, lattice paths, and perfect matchings and the weak Lefschetz property
• Secant Varieties of the Varieties of Reducible Hypersurfaces in ${\mathbb{P}}^n$
• A classification of the weak Lefschetz property for almost complete intersections generated by uniform powers of general linear forms
• The Limiting Polytope of the Generic Initial System of a Complete Intersection
• Symmetry, Unimodality, and Lefschetz Properties for Graded Modules
• The EGH Conjecture and the Sperner property of complete intersections
• On the Weak-Lefschetz property for Artinian Gorenstein algebras