Geometric meaning of the residual irreducible polynomials

Identify the geometric meaning of the irreducible polynomial factors remaining after extracting the shifted-product factors from the incidence-degree formulas for involutive cone loci in projective symplectic spaces.

Background

The paper derives Segre-class formulas for the degrees of loci of hypersurfaces containing involutive cones. In several cases, these degree polynomials factor into explicit shifted binomial factors and an additional irreducible polynomial, such as the degree-36 polynomial $P_{36}(d)$ for quadric cones in $\mathbb{P}^5$.

The shifted factors are explained through the structure of the Segre-class coefficient formula and vanishing at special values of the degree parameter. However, the remaining irreducible factors are not given a geometric interpretation, leaving open the problem of explaining what geometric enumerative phenomena they encode.