A tour of the Weak and Strong Lefschetz Properties

TL;DR

This paper surveys the development and applications of the Weak and Strong Lefschetz Properties in algebraic geometry, highlighting key theorems, research directions, and open questions in the field.

Contribution

It provides a comprehensive overview of the evolution of Lefschetz properties, emphasizing their foundational theorem and current research challenges.

Findings

The Weak Lefschetz Property is central to various algebraic and geometric theories.

Research has expanded from Stanley's theorem to diverse applications and open problems.

The paper identifies ongoing questions motivating future investigations.

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.