The weak Lefschetz property, monomial ideals, and lozenges

摘要

We study the weak Lefschetz property and the Hilbert function of levelArtinian monomial almost complete intersections in three variables. Severalsuch families are shown to have the weak Lefschetz property if thecharacteristic of the base field is zero or greater than the maximal degree ofany minimal generator of the ideal. Two of the families have an interestingrelation to tilings of hexagons by lozenges. This lends further evidence to aconjecture by Migliore, Miro-Roig, and the second author. Finally, using ourresults about the weak Lefschetz property, we show that the Hilbert function ofeach level Artinian monomial almost complete intersection in three variables ispeaked strictly unimodal.