Lefschetz properties in higher codimensions for binomial Macaulay dual generators

Determine explicit conditions on the exponents a_1,…,a_n and b_1,…,b_n in the homogeneous binomial Macaulay dual generator F = X_1^{a_1}⋯X_n^{a_n}(X_1^{b_1}⋯X_r^{b_r} − X_{r+1}^{b_{r+1}}⋯X_n^{b_n}) with 1 ≤ r ≤ n − 1, that guarantee the associated Artinian Gorenstein algebra A_F over a field of characteristic zero has the weak Lefschetz property or the strong Lefschetz property in codimension n ≥ 4.

Background

The paper studies Artinian Gorenstein algebras A_F whose Macaulay dual generator F is a binomial. Monomial complete intersections are known to have both WLP and SLP in characteristic zero, and the companion work establishes these properties in codimension three for binomial generators. However, examples in codimension four show that WLP may fail, indicating that a full characterization in higher codimensions is nontrivial.

This work provides several new families of binomial Macaulay dual generators that ensure WLP or SLP and develops tools via Hessians and connected sums, but a general set of conditions covering higher codimensions remains to be identified.