Related papers: Realization spaces of matroids over hyperfields
In a recent paper Baker and Bowler introduced matroids over hyperfields, offering a common generalization of matroids, oriented matroids, and linear subspaces of based vector spaces. This paper introduces the notion of a topological…
A phased matroid is a matroid with additional structure which plays the same role for complex vector arrangements that oriented matroids play for real vector arrangements. The realization space of an oriented (resp., phased) matroid is the…
We present an algebraic framework which simultaneously generalizes the notion of linear subspaces, matroids, valuated matroids, and oriented matroids. We call the resulting objects matroids over hyperfields. In fact, there are (at least)…
We introduce a new model for the realization space of a matroid, which is obtained from a variety defined by a saturated determinantal ideal, called the slack ideal, coming from the vertex-hyperplane incidence matrix of the matroid. This is…
To every realizable oriented matroid there corresponds an arrangement of real hyperplanes. The homeomorphism type of the complexified complement of such an arrangement is completely determined by the oriented matroid. In this paper we study…
A matroid $M$ is an ordered pair $(E,I)$, where $E$ is a finite set called the ground set and a collection $I\subset 2^{E}$ called the independent sets which satisfy the conditions: (i) $\emptyset \in I$, (ii) $I'\subset I \in I$ implies…
We study the realization spaces of matroids and hyperplane arrangements. First, we define the notion of naive dimension for the realization space of matroids and compare it with the expected dimension and the algebraic dimension, exploring…
A matroid is a machine capturing linearity of mathematical objects and producing combinatorial structures. Matroid structure arises everywhere since linearity is a ubiquitous concept. One natural way to obtain matroids is by considering…
This article is a survey of matroid theory aimed at algebraic geometers. Matroids are combinatorial abstractions of linear subspaces and hyperplane arrangements. Not all matroids come from linear subspaces; those that do are said to be…
Based on the notion of vectors and linear subspaces for a matroid, we develop a theory of flats and hyperplane arrangements for T-matroids, where T is a tract. This leads to several cryptomorphic descriptions of T-matroids: in terms of its…
We present an algebraic framework which simultaneously generalizes the notion of linear subspaces, matroids, valuated matroids, oriented matroids, and regular matroids. To do this, we first introduce algebraic objects called tracts which…
Baker and Bowler showed that the Grassmannian can be defined over a tract, a field-like structure generalizing both partial fields and hyperfields. This notion unifies theories of matroids over partial fields, valuated matroids, and…
It is known that there exist hyperplane arrangements with same underlying matroid that admit non-homotopy equivalent complement manifolds. In this work we show that, in any rank, complex central hyperplane arrangements with up to 7…
We introduce the notion of a matroid M over a commutative ring R, assigning to every subset of the ground set an R-module according to some axioms. When R is a field, we recover matroids. When R=$\mathbb{Z}$, and when R is a DVR, we get…
We introduce combinatorial types of arrangements of convex bodies, extending order types of point sets to arrangements of convex bodies, and study their realization spaces. Our main results witness a trade-off between the combinatorial…
Swartz proved that any matroid can be realized as the intersection lattice of an arrangement of codimension one homotopy spheres on a sphere. This was an unexpected extension from the oriented matroid case, but unfortunately the…
We study smoothness of realization spaces of matroids for small rank and ground set. For $\mathbb{C}$-realizable matroids, when the rank is $3$, we prove that the realization spaces are all smooth when the ground set has $11$ or fewer…
Every bi-uniform matroid is representable over all sufficiently large fields. But it is not known exactly over which finite fields they are representable, and the existence of efficient methods to find a representation for every given…
We explore a combinatorial theory of linear dependency in complex space, "complex matroids", with foundations analogous to those for oriented matroids. We give multiple equivalent axiomatizations of complex matroids, showing that this…
We extend vector configurations to more general objects that have nicer combinatorial and topological properties, called weighted pseudosphere arrangements. These are defined as a weighted variant of arrangements of pseudospheres, as in the…