Lefschetz properties for monomial ideals

Abstract

This work is centred on studying a conjecture affirming the existence of monomial ideals satisfying the weak Lefschetz property for any number of generators. We prove the conjecture for three variables, and show that it also holds in any number of variables when the number of generators is big enough. We also show that for a higher number of variables, the problem can be reduced to a problem of the strong Lefschetz property in one less variable.

Notation: In this work $K$ will denote a characteristic zero field, $R$ the polynomial ring $K[x_{1},\ldots,x_{n}]$, and $\mathfrak{m}$ the maximal ideal $(x_{1},\ldots,x_{n})$.