Algebra/Topology Seminar by Simona Settepanella

Talk by Simona Settepanella (Hokkaido University)

Title: Intersection lattice of Discriminantal arrangement and hypersurfaces in Grassmannian.

Abstract: In 1989 Manin and Schechtman considered a family of arrangements of hyperplanes generalizing classical braid arrangements which they called the Discriminantal arrangements. Such an arrangement consists of parallel translates of collection of n hyperplanes in general position in C^k which fail to form a generic arrangement.

In 1994 Falk showed that the combinatorial type of Discriminantal arrangement depends on the collection of n hyperplanes in general position in C^k. In 1997 Bayer and Brandt divided generic arrangements in C^k in "very generic" and "non very generic" depending of the intersection lattice of associated Discriminatal arrangement. In 1999 Athanasiadis provided a full description of intersection lattice for Discriminantal arrangement in the very generic case. More recently, in 2016, Libgober and Settepanella gave a description of rank 2 intersection lattice of Discriminantal arrangement in non very generic case, providing a sufficient geometric condition for a generic arrangement in C^k to be non very generic. In this talk we will recall their result and we will show that non very generic arrangements in C^3 satisfying their condition correspond to points in a degree 2 hypersurface in the complex Grassmannian Gr(3,n). This is a joint work with S. Sawada and S. Yamagata.