Matthew Davidson Booth: Further Progress on the Weak Lefschetz Property for Monomial Almost Complete Intersections in Three Variables

Abstract: Much progress has been made in classifying when the weak Lefschetz property (WLP) holds for the standard graded Artinian algebra \(A=\mathbb{F}[x,y,z]/I\) where \(\operatorname{char}(\mathbb{F})=0\) and \(I=(x_{1}^{d_{1}},y^{d_{2}},z^{d_{3}},x^{a_{1}}y^{a_{2}}z^{a_{3}})\) is a monomial almost complete intersection. In particular, a conjecture made in 2011 by Migliore, Miró-Roig, and Nagel has largely been verified in numerous incremental steps, culminating with a theorem in 2021 due to Cook and Nagel. To handle some of the remaining open cases, we connect the problem of deciding WLP in this three-variable setting to the setting of two variables through a certain relation. In so doing, we are led to determine explicit formulas for the generators of the colon ideal \((x^{d_{1}},y^{d_{2}}):(x+y)^{a_{3}}\). With these generators in hand, we construct a matrix and show that failure of WLP for A is dictated by the vanishing of a certain polynomial (namely the determinant of our matrix) when \(A\) is level. This is joint work with Adela Vraciu.