I'm making my way through Richard P. Stanley's An Introduction to Hyperplane Arrangements, and am having a bit of difficulty processing the definition of triple's of arrangements. The document states, on page 13 that:
2.1. Properties of the intersection poset.
Let $\mathcal{A}$ be an arrangement in the vector space V. A subarrangement of $\mathcal{A}$ is a subset $\mathcal{B}$ ⊆ $\mathcal{A}$. Thus $\mathcal{B}$ is also an arrangement in V. If $x$ ∈ L($\mathcal{A}$), define the subarrangement $\mathcal{A}_x$ ⊆ $\mathcal{A}$ by
$\mathcal{A}_x$ = {$H \in \mathcal{A} : x \subseteq H$}.
Also define an arrangement $\mathcal{A}^x$ in the affine subspace x ∈ L($\mathcal{A}$) by
$\mathcal{A}^x$ ={$x$∩H$\neq$∅ : $H \in \mathcal{A} - \mathcal{A}_x$}.
Where $L(\mathcal{A})$ is the poset of $\mathcal{A}$. Now let me first say that I believe the definition of $\mathcal{A}^x$ to be slightly incorrect, as it should be an arrangement, and should instead read:
$\mathcal{A}^x = \{H \in \mathcal{A} - \mathcal{A}_x: x \cap H\neq \emptyset\}$.
That is, all hyperplanes that intersect $x$ but are not supersets of $x$. A little bit further down on page 13 it states:
Choose $H_0 \in \mathcal{A}$. Let $\mathcal{A}′$ = $\mathcal{A}−\{H_0\}$ and $\mathcal{A}′′$ = $A^{H_0}$. We call $(\mathcal{A},\mathcal{A}′,\mathcal{A}′′)$ a triple of arrangements with distinguished hyperplane $H_0$.
An example figure is provided on the following page:
I can easily see how $\mathcal{A}$ and $\mathcal{A}'$ match the definitions provided. However, my understanding of $\mathcal{A}''$ is:
$\mathcal{A}'' = \mathcal{A}^{H_0} = \{H \in \mathcal{A} - \mathcal{A}_{H_0}: H_0 \cap H\neq \emptyset\}$.
where
$\mathcal{A}_{H_0} = \{ H_0 \}$.
With respect to the diagram, and if my understanding is correct, I make 3 points:
- $H_0$ should not be part of the diagram, but obviously is.
- There are 3 hyperplanes that should be part of the solution, but aren't.
- There seems to be emphasis on points of intersection that is not part of the definition.
Going by the figure, it strikes me that maybe the definition of $\mathcal{A}^x$ is meant to be:
$\mathcal{A}^x = \{y: y \in L(\mathcal{A}), y \subseteq x\}$
But obviously this is nothing like the given definition. Could someone clarify the explanation please?
I have also checked the errata document that can be found here, but that does not provide any additional clarity.