Star configurations in \(\mathbb{P}^n\) (Q360180)

A \textit{star configuration} of codimension \(c\), \(V_c\subset \mathbb{P}^n\), is defined as follows: let \(\mathcal{H} = (H_1,\dots,H_s)\) be a set of hyperplanes in \(\mathbb{P}^n\) such that they intersect properly, i.e. the intersection of \(j\) elements of \(\mathcal{H}\) has codimension \(j\); we define \(V_c = V_c(\mathcal{H} , \mathbb{P}^n) = \bigcup_{1\leq i_1\leq \dots \leq i_c\leq s} H_{i_1}\cap \dots\cap H_{i_c}\).NEWLINENEWLINELet \(L_i\) be the defining linear form of \(H_i\), then \(I_{V_c} = \bigcap _{1\leq i_1\leq \dots \leq i_c\leq s} (L_{i_1},\dots L_{i_c}).\)NEWLINENEWLINEThe study of star configurations is related to several problems such as determining the Hilbert function for points in \(\mathbb{P}^2\) or the resurgence for certain projective schemes, and the study of secant varieties of some classical varieties.NEWLINENEWLINEThe main results in the paper are that not only every star configuration is arithmetically Cohen-Macaulay (ACM), but also every symbolic power \(I_{V_c}^{(m)}\) of the ideal of a star configuration defines an ACM scheme in \( \mathbb{P}^n\); moreover for certain positive dimensional star configurations, the \textit{resurgence} \(\rho(I_{V_c}) = \sup \{\frac{m}{r}: I_{V_c}^{(m)} \nsubseteq I_{V_c}^r\} \) of their ideal is determined, one of the few examples that can be found in the literature.