Geometric Equations for Matroid Varieties
| Autor: | Ashley Wheeler, Jessica Sidman, Will Traves |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2019 |
| Předmět: |
010102 general mathematics
14M15 05B35 0102 computer and information sciences 01 natural sciences Matroid Stratification (mathematics) Theoretical Computer Science Combinatorics Mathematics - Algebraic Geometry Computational Theory and Mathematics 010201 computation theory & mathematics FOS: Mathematics Discrete Mathematics and Combinatorics Mathematics - Combinatorics Gravitational singularity Combinatorics (math.CO) 0101 mathematics Algebraic Geometry (math.AG) Mathematics Projective geometry |
| Popis: | 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 $\{y \in \mathrm{Gr}(r,n) | M_y = M_x\}$ 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. Comment: Updated Proposition 2.1.3. Added Theorem 2.1.7 and Remark 3.0.3 |
| Databáze: | OpenAIRE |
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