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Register a new userEfficient computation of resonance varieties via Grassmannians
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Associated to the cohomology ring A of the complement X(A) of a hyperplane arrangement A in complex m-space are the resonance varieties R^k(A). The most studied of these is R^1(A), which is the union of the tangent cones at the origin to the characteristic varieties of the fundamental group of X. R^1(A) may be described in terms of Fitting ideals, or as the locus where a certain Ext module is supported. Both these descriptions give obvious algorithms for computation. In this note, we show that interpreting R^1(A) as the locus of decomposable two-tensors in the Orlik-Solomon ideal leads to a description of R^1(A) as the intersection of a Grassmannian with a linear space, determined by the quadratic generators of the Orlik-Solomon ideal. This method is much faster than previous alternatives.
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In this article, we present a massively parallel framework for computing tropicalizations of algebraic varieties which can make use of finite symmetries. We compute the tropical Grassmannian TGr$_0(3,8)$, and show that it refines the $15$-dimensional skeleton of the Dressian Dr$(3,8)$ with the exception of $23$ special cones for which we construct explicit obstructions to the realizability of their tropical linear spaces. Moreover, we propose algorithms for identifying maximal-dimensional tropical cones which belong to the positive tropicalization. These algorithms exploit symmetries of the tropical variety even though the positive tropicalization need not be symmetric. We compute the maximal-dimensional cones of the positive Grassmannian TGr$^+(3,8)$ and compare them to the cluster complex of the classical Grassmannian Gr$(3,8)$.
We study conditional independence (CI) models in statistical theory, in the case of discrete random variables, from the point of view of algebraic geometry and matroid theory. Any CI model with hidden random variables corresponds to a variety defined by certain determinantal conditions on a matrix whose entries are probabilities of events involving the observed random variables. We show that any CI variety, and more generally any hypergraph variety, admits a matroid stratification. Our main motivation for studying decompositions of CI varieties is the realizability problem: given a collection of CI relations, the goal is to determine the existence of random variables that satisfy these constraints and violates the rest. We show that the realization spaces of CI models and the matroid varieties in their decompositions are closely related. We use ideas from incidence geometry, in particular point and line configurations, to find minimal decompositions of general hypergraph varieties in terms of matroid varieties, which are not necessarily irreducible by Mnev--Sturmfels universality theorem, and may have arbitrary singularities. We focus on various families of hypergraph varieties for which we explicitly compute an irredundant irreducible decomposition. Our main findings in this direction are threefold: (1) we describe minimal matroids of such hypergraphs; (2) we prove that the varieties of these matroids are irreducible and their union is the hypergraph variety; and (3) we show that every such matroid is realizable over real numbers. Our decomposition strategy gives immediate matroid interpretations of the irreducible components of many families of CI varieties in algebraic statistics, and unravels the symmetric structures in CI varieties which hugely simplifies the computations.
The Springer variety is the set of flags stabilized by a nilpotent operator. In 1976, T.A. Springer observed that this varietys cohomology ring carries a symmetric group action, and he offered a deep geometric construction of this action. Sixteen years later, Garsia and Procesi made Springers work more transparent and accessible by presenting the cohomology ring as a graded quotient of a polynomial ring. They combinatorially describe an explicit basis for this quotient. The goal of this paper is to generalize their work. Our main result deepens their analysis of Springer varieties and extends it to a family of varieties called Hessenberg varieties, a two-parameter generalization of Springer varieties. Little is known about their cohomology. For the class of regular nilpotent Hessenberg varieties, we conjecture a quotient presentation for the cohomology ring and exhibit an explicit basis. Tantalizing new evidence supports our conjecture for a subclass of regular nilpotent varieties called Peterson varieties.
We classify complex projective varieties of dimension $2r geq 8$ swept out by a family of codimension two grassmannians of lines $mathbb{G}(1,r)$. They are either fibrations onto normal surfaces such that the general fibers are isomorphic to $G(1,r)$ or the grassmannian $mathbb{G}(1,r+1)$. The cases $r=2$ and $r=3$ are also considered in the more general context of varieties swept out by codimension two linear spaces or quadrics.
We study standard monomial bases for Richardson varieties inside the flag variety. In general, writing down a standard monomial basis for a Richardson variety can be challenging, as it involves computing so-called defining chains or key tableaux. However, for a certain family of Richardson varieties, indexed by compatible permutations, we provide a very direct and straightforward combinatorial rule for writing down a standard monomial basis. We apply this result to the study of toric degenerations of Richardson varieties. In particular, we provide a new family of toric degenerations of Richardson varieties inside flag varieties.
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