Tope Arrangements and Determinantal Varieties

Tope arrangements are a collection of bipartite graphs on node sets $[n]\sqcup[d]$, along with a bijection to the lattice points of the scaled simplex $n\Delta_{d-1}$. The motivation for their introduction arises from tropical geometry, but they have multiple connections to classical algebraic geometry via the Grassmannian $\rm{Gr}_d(\mathbb{C}^n)$ and the determinantal variety $\nabla_{d,n}$, the variety of degenerate matrices: \[ \nabla_{d,n} = \left\{X \in \mathbb{C}^{d\times n}\,\right|\left.\,rk(X) < d\right\} \enspace . \] In this talk, we shall focus on the connections between the latter object and tope arrangements. Furthermore, we will show how this combinatorial machinery solves two conjectures of Sturmfels and Zelevinsky regarding determinantal varieties and their Chow forms.

This is joint work with Georg Loho.