Discrete Geometric Structure Seminar
Discriminantal Arrangement, 3×3 Minors of Plücker Matrix and Hypersurfaces in Grassmannian Gr(3,n)
So Yamagata (Hokkaido University)
2018/01/25 (Thu) 11:00-11:20 (Rm 3-413)
2018/01/25 (Thu) 11:00-11:20 (Rm 3-413)
For a generic arrangement of n hyperplanes A = { H^0_1, H^0_2, ..., H^0_n } in C^k, k<n consider the space S ≅ C^n of n-tuples (H^{t_1}_1, H^{t_2}_2, ..., H^{t_n}_n), t_i ∈ C of parallel translates of (H^0_1, H^0_2, ..., H^0_n).
The closed subset of S formed by translates of hyperplanes in A that fail to form a generic arrangement defines an arrangement of hyperplanes. This arrangement B(n,k) is defined by Manin and Schechtman in 1989 and called the discriminantal arrangement.
In 1999 Falk showed that the combinatorial type of B(n,k) depends on original arrangement by providing an example. In 1997 Bayer and Brandt called an arrangement A very generic if for any rank the cardinality of intersection lattice of B(n, k) is the maximum possible number and non very generic otherwise. They conjectured a description of combinatorics of discriminantal arrangement associated to very generic arrangement. In 1999 Athanasiadis proved their conjecture and so a full description of discriminantal arrangement for very generic arrangement was completed. However for non very generic arrangements description of combinatorics of discriminantal arrangement is only known for rank 2 intersections (by a work of Libgober and Settepanella).
In 2017 Sawada, Settepanella and Yamagata applying result of Libgober and Settepanella proved that polynomial p_T(a_{ij}) introduced by Athanasiadis to prove the conjecture by Bayer and Brandt has a simpler polynomial expression if T is a set of cardinality 3. Moreover they proved that non very generic arrangements of n hyperplanes in C^3 are points in a hypersurface in complex Grassmannian Gr(3,n). They also showed that this hypersurface is union of quadrics of which they provided full description. This thesis is written based on their results.