Open Collections

Description

Let $R$ be a polynomial ring in $r$ variables. Let $L$ be a general linear form. A result of Stanley and of Watanabe says that if I is an ideal generated by $r$ powers of linearly independent linear forms (e.g. the variables), and if $j$ and $k$ are any positive integers, then multiplication by $L^k$ from the degree $j$ component of $R/I$ to the degree $j+k$ component has maximal rank. This leads to the question of what happens when we have powers of more than $r$ linear forms. Over the last decade or so several papers have addressed different aspects of this problem, most often focusing on the case $k=1$ (i.e. studying the so-called Weak Lefschetz Property). In the last half year or so, progress has been made on the problem when $k > 1$. I will give an overview of the history of this problem, talk about recent work in collaboration with R. Miró-Roig and with U. Nagel, and mention open problems and directions for new research.