At the moment, I am reading the paper "on the connectivity of the realization spaces of line arrangements" of Nazir and Yoshinaga.

I would like to extend their Lemma 3.2 to higher dimension. However, I need to better understand the notion of hyperplane generic to a given arrangement.

Question 1: Let $\mathcal{A}=\{H_{1},\ldots,H_{m}\}$ be a hyperplane arrangement in $\mathbb{CP}^{d}$ and let $K$ be a hyperplane of $\mathbb{CP}^{d}$. What does it mean that $K$ is generic with respect to $\mathcal{A}$?

Question 2: For $d=2$ (the case of line arrangements) this notion is equivalent to say that $\det(H_{i},H_{j},K)\neq0$ for $1\leq i<j\leq m$ (here we are assuming that the lines of $\mathcal{A}$ are pairwise distinct). How can we generalize this characterization to higher dimension?


1) This means that any intersections of $K$ with the hyperplane system is of the dimension, that you might expect. So the intersection of $K$ and $n$ of the hypersurfaces $H_i$ is of codimension $n+1$ for $n<d$ and empty for higher $n$.

2) I'm not sure if I understand this determinant correctly. You could look at the matrix built by the normal vectors of these hyperplanes ($n_i$ of the $H_i$, $n_K$ of $K$), seen as hyperplanes in $\mathbb{C}^{d+1}$. If $m<d$, you can say that $K$ is in general position with respect to the $H_i$ if this matrix $(n_1,\ldots,n_m,n_K)$ has maximal rank, i.e. $rank(n_1,\ldots,n_m)+1$.


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