Secant variety to a zero-dimensional projective variety

Question

This is a reference request/nomenclature question. Let $A \subseteq \mathbb{P}^n$ be a finite set of points not contained in a hyperplane (over some field), and let $\sigma_r(A)$ be the $r$-th secant variety to $A$. This secant variety forms a subspace arrangement, i.e., a finite union of linear subspaces of $\mathbb{P}^n$. Is there a specific name for subspace arrangements of this form? Surely such arrangements have been studied, and I would be very grateful for a reference.

Answer 1

I believe this would be a dual arrangement of a star arrangement.

A star arrangement is a union of subspaces defined as follows. Let $H_1,\dotsc,H_d$ be a collection of hyperplanes and fix an integer $c$. The codimension $c$ star arrangement $X_c$ is the union of intersections of $c$ of the $H_i$, over all size $c$ subsets of $H_1,\dotsc,H_d$. Usually there is some hypothesis of linear generality so that any $c$ of the $H_i$'s are independent. See for example https://arxiv.org/abs/1203.5685, https://arxiv.org/abs/1801.04579.

Given a subspace arrangement $\mathcal{A} = \{W_1,\dotsc,W_s\}$ of subspaces in $\mathbb{P}V$, the dual arrangement is $\mathcal{A}^* = \{W_1^\perp,\dotsc,W_s^\perp\}$ in $\mathbb{P}V^*$.

Well, the subspace $W = H_{i_1} \cap \dotsb \cap H_{i_c}$ is dual to $W^\perp = \operatorname{span}\{H_{i_1}^\perp,\dotsc,H_{i_c}^\perp\}$. So the secant varieties you're asking about are dual arrangements of star arrangements. I don't know a better or alternative name and I'm not aware of any work on these arrangements specifically, but the authors of the papers I linked might be able to give more information if you write to them.

By the way, these dual arrangements of star arrangements are not necessarily themselves star arrangements. Let $A$ be a set of $5$ general points in $\mathbb{P}^3$ so $\sigma_2(A)$ is a set of $10$ lines. This arrangement of lines has $4$ through each node (point where lines meet) and $3$ in each plane spanned by two meeting lines. This arrangement of lines isn't a star arrangement. A star arrangement given by pairwise intersections of $5$ general hyperplanes would again consist of $10$ lines, but this time with $4$ in each plane spanned by two meeting lines, and $3$ lines through each node.