Dimension bound for adjoints of generic positive hexahedra

Establish that a generic positive hexahedron in the Grassmannian \(\mathrm{Gr}(2,4)\) has at most three linearly independent adjoint polynomials modulo the Plücker quadric defining \(\mathrm{Gr}(2,4)\).

Background

Positive hexahedra in Gr(2,4)\mathrm{Gr}(2,4) are obtained by intersecting the positive Grassmannian with two additional half-spaces. Their adjoints are quadratic hypersurfaces in the Plücker projective space. The paper proves that, for generic choices of the two additional hyperplanes, there are at most six linearly independent adjoints modulo the Plücker quadric. The authors report that all computed examples have adjoint families of dimension at most three and conjecture that this smaller bound holds generically.

References

We conjecture that this family is in fact at most three-dimensional and show on an example that this bound is attained.

— Positive Polytopes with Few Facets in the Grassmannian  (2503.01652 - Pavlov et al., 3 Mar 2025) in Introduction; Conjecture in Section 4, subsection “Hexahedra”