Generic reverse lexicographic property for comaximal determinantal ideals

Establish that for any integer n ≥ 3, with k = 4 variables, r = n − 2, and d = 1 (i.e., for the ideal I_{n−2}(M) generated by all (n−1)-minors of an n × n matrix M of homogeneous linear forms in four variables), there exists a nonempty Zariski open subset U ⊂ k^{4n^2} such that for all specializations a ∈ U, the determinantal ideal I_{n−2}(φ_a(𝔐)) is reverse lexicographic with respect to the graded reverse lexicographic order.

Background

The paper introduces the property RL(I), which asserts that a homogeneous ideal I ⊂ k[x1,x2,x3,x4] is reverse lexicographic, meaning its grevlex staircase is closed upward among monomials of the same degree. The authors focus on the comaximal determinantal case for square matrices of linear forms with k = 4 variables, r = n − 2, and d = 1, yielding a zero-dimensional ideal I_{n−2}(M).

They conjecture that this reverse lexicographic property holds generically for I_{n−2}(M) under specialization from a generic n × n matrix of linear forms, which would enable the structural analysis of grevlex staircases and the sharp complexity bounds for the DetGB algorithm. The conjecture is connected to Lefschetz properties, particularly the 2-strong Lefschetz property, which the authors discuss as a possible route to proving generic reverse lexicographic behavior.