2025/03/03 by Д. С. Павлов, Kristian Ranestad, Pavlov, Dmitrii +1 · 1 citation
Mathematics · #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics #Mathematics and Applications #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.2503.01652
openalex publication_date 2025/03/03 · openalex created_date 2025/10/12 · openalex updated_date 2026/07/28
In this article we study adjoint hypersurfaces of geometric objects obtained by intersecting simple polytopes with few facets in ℙ5 with the Grassmannian Gr(2,4). These generalize the positive Grassmannian, which is the intersection of Gr(2,4) with the simplex. We show that if the resulting object has five facets, it is a positive geometry and the adjoint hypersurface is unique. For the case of six facets we show that the adjoint hypersurface is not necessarily unique and give an upper bound on the dimension of the family of adjoints. We illustrate our results with a range of examples. In particular, we show that even if the adjoint is not unique, a positive hexahedron can still be a positive geometry.