Geometric equations for matroid varieties

Algebra Seminar
Tuesday, October 5, 2021 - 10:00am for 1 hour (actually 50 minutes)
Skiles 006
Ashley K. Wheeler – Georgia Tech –
Ashley K. Wheeler

Each point x in Gr(r, n) corresponds to an r × n matrix A_x which gives rise to a matroid M_x on its columns. Gel’fand, Goresky, MacPherson, and Serganova showed that the sets {y ∈ Gr(r, n)|M_y = M_x} form a stratification of Gr(r, n) with many beautiful properties. However, results of Mnëv and Sturmfels show that these strata can be quite complicated, and in particular may have arbitrary singularities. We study the ideals I_x of matroid varieties, the Zariski closures of these strata. We construct several classes of examples based on theorems from projective geometry and describe how the GrassmannCayley algebra may be used to derive non-trivial elements of I_x geometrically when the combinatorics of the matroid is sufficiently rich.