# Tagged Questions

A hyperplane arrangement is a set of hyperplanes in a vector space or in a projective space. The complement of the union of these hyperplanes defines an algebraic variety, with interesting geometry and topology.

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### Counting problems where unlabeled is easier than labeled

I was encouraged to post this question by Jim Propp during a meeting of the Cambridge Combinatorics and Coffee Club. It is a counterpoint to the MathOverflow question "Counting Problems where Labeled ...
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### Characterizing normal vectors of affinely independent subsets of the hypercube

Suppose we are looking at the hypercube in $\mathbb{R}^n$. I tend to the think of the cube with vertex coordinates 0 or 1, but maybe this is easier for the $\pm1$ cube. Now suppose we have an affine ...
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### Maximal Number of Pairs of Orthogonal vectors in a set of $n$ vectors in $\mathbb{R}^3$

Suppose you are given a set of $n$ non-zero vectors in $\mathbb{R}^3$. What is the maximum number of pairs of them that are orthogonal? The current guess is $\le 2n$. EDIT: I forgot to add that no ...
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### Details on the Symmetric Group action on chambers of the Shi Arrangement

In "A Simple Bijection for the Regions of the Shi Arrangement of Hyperplanes", http://arxiv.org/pdf/math/9702224v1.pdf, Athanasiadis and Linusson give a bijection between the regions of the Shi ...
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### Open problems in hyperplane/subspace arrangements?

What are some open problems in hyperplane/subspace arrangements, preferably of the combinatorial algebraic topology kind, and where can one read about them? That is, where are they discussed, and ...
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### Volume of bounded regions in hyperplane arrangements

I am given a hyperplane arrangement $\mathcal{H}_0$ in $\mathbb{R}^n$ and a function $\phi \colon \mathbb{R}^n \to \mathbb{Q}.$ I choose any enumeration on the set of primitive vectors (i.e. vectors ...
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### What are / could be the applications of Delaunay oriented matroids?

The Delaunay oriented matroid is studied in detail by F. Santos (actually this paper is more general). For a set of points $S$ (in any dimension), let $C$ be a sphere, $C^+$ be its interior, $C^-$ be ...
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### help with cohomology of $\mathbb{P}^n$ relative to a NCD

Let $H_0, \ldots, H_n$ be $n$ hyperplanes in $\mathbb{P}^n(\mathbb{C})$ with normal crossings and denote by $H$ the union of them. I am trying to understand why (1) $H^n(\mathbb{P}^n(\mathbb{C}), H)$...
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Let $H$ be a set of $(d-1)$-dimensional hyperplanes in $\mathbb{R}^d$. For each hyperplane $h \in H$ let $D(h)$ and $\bar{D}(h)$ be the corresponding half spaces of $\mathbb{R}^d$. For a point $x \... 0answers 505 views ### Maximal disjoint hyperplanes Assume a set of$n^{r}$points$X_{r} = \{ x_{1}, \cdots, x_{n^{r}} \}$is given occupying a codimension$t^{r}$subspace in$\mathbb{R}^{n^{r}}$. Let$M_{r}$be the set of$t^{r}$-tuples of these ... 1answer 652 views ### A natural refinement of the$A_n$arrangement is to consider all$2^n-1$hyperplanes given by the sums of the coordinate functions. Have you seen this arrangement? Is it completely intractable? The short version Here is an extremely natural hyperplane arrangement in$\mathbb{R}^n$, which I will call$R_n$for resonance arrangement. Let$x_i$be the standard coordinates on$\mathbb{R}^n$. ... 0answers 233 views ### balls in arrangements of hyperplanes The following theorem is from Aronov, Naiman, Pach and Sharir's An invariant property of balls in arrangements of hyperplanes. I would like to state them and then ask if any related problem/theorem ... 1answer 242 views ### Criterion for being a simple vector 1) I was wondering if there exists a criterion (of (a) combinatorial or (b) geometric nature) for a sum of simple vectors$V\in\wedge^k(\mathbb R^n)$,$V=e_{a_{11}}\wedge\cdots\wedge e_{a_{1k}} + \...
Let $\{A_i\}$ be a collection of $m$ hyperplanes in $\mathbb{C}^n$ which all pass through the origin (a central hyperplane arrangement). Such an arrangement is called Coxeter if reflecting across any ...
Let $F= \frac{G}{H} : \mathbb P^n \to \mathbb P^1$ be a non-constant rational function ($G$ and $H$ homogenous polynomials of the same degree in $\mathbb C^{n+1})$. The fiber over \$(\lambda:\mu) \in ...