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Abstract Polymatroids are combinatorial abstractions of subspace arrangements in the same way that matroids are combinatorial abstractions of hyperplane arrangements. By introducing augmented Chow rings of polymatroids, modeled after augmented wonderful varieties of subspace arrangements, we generalize several algebro-geometric techniques developed in recent years to study matroids. We show that intersection numbers in the augmented Chow ring of a polymatroid are determined by a matching property known as the Hall–Rado condition, which is new even in the case of matroids.
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Intersection Theory of Matroids: Variations on a Theme
Chow rings of toric varieties, which originate in intersection theory, feature a rich combinatorial structure of independent interest. We survey four different ways of computing in these rings, due to Billera, Brion, Fulton–Sturmfels, and Allermann–Rau. We illustrate the beauty and power of these methods by giving four proofs of Huh and Huh–Katz’s formula μ_k(M) = deg_M(α^{r−k}β^k) for the coefficients of the reduced characteristic polynomial of a matroid M as the mixed intersection numbers of the hyperplane and reciprocal hyperplane classes α and β in the Chow ring of M. Each of these proofs sheds light on a different aspect of matroid combinatorics, and provides a framework for further developments in the intersection theory of matroids. Our presentation is combinatorial, and does not assume previous knowledge of toric varieties, Chow rings, or intersection theory. This survey was prepared for the Clay Lecture to be delivered at the 2024 British Combinatorics Conference.
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- Award ID(s):
- 2154279
- PAR ID:
- 10587894
- Publisher / Repository:
- Cambridge University Press
- Date Published:
- ISBN:
- 9781009490535
- Page Range / eLocation ID:
- 1 to 30
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Book Chapter:
- https://doi.org/10.1017/9781009490559.002
