MATH Seminar
| Title: Geometric equations for matroid varieties |
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| Seminar: Algebra and Number Theory |
| Speaker: Ashley Wheeler of Georgia Institute of Technology |
| Contact: David Zureick-Brown, dzureic@emory.edu |
| Date: 2021-10-05 at 4:00PM |
| Venue: MSC W301 |
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| Abstract: Each point $x$ in $Gr(r, n)$ corresponds to an $r \times n$ matrix $A_x$ which gives rise to a matroid $M_x$ on its columns. Gel'fand, Goresky, MacPherson, and Serganova showed that the sets $\{y \in Gr(r, n)\,|\,M_y = M_x\}$ form a stratification of $Gr(r, n)$ with many beautiful properties. However, results of Mn\"ev 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 Grassmann--Cayley algebra may be used to derive non-trivial elements of $I_x$ geometrically when the combinatorics of the matroid is sufficiently rich. |
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