Inequalities for free multi-braid arrangements. (English) Zbl 1403.13040

Let \(V\cong\mathbb K^{l+1}\) be a vector space over a field \(\mathbb K\) of characteristic zero, \(V^\ast\) its dual space and \(S = \mathrm{Sym}(V^\ast)\cong\mathbb K[x_0,\dots, x_l]\). For \(0\leq i<j\leq l\), let \(\alpha_{ij}=x_i-x_j\) and let \(H_{ij}\) denote the corresponding hyperplane. The braid arrangement of type \(A_l\subset V\) is defined as \(\bigcup_{0\leq i < j\leq l}H_{ij}\). A multiplicity on \(A_l\) is a map \(\mathfrak m:\{H_{ij}\}\rightarrow\mathbb N\); it is represented as the tuple \((m_{ij}=\mathfrak m(H_{ij})\,)\) in the multiplicity lattice \(\Lambda_l=\mathbb N^{{l+1}\choose 2}\); \(m_{ij} = m_{ji}\) for any \(i, j\). The pair \((A_l,\mathfrak m)\) is called a multi-braid arrangement; it is free if the module \(D(A_l,\mathfrak m) = \{\theta\in\mathrm{Der}(S)\mid\theta(\alpha_{ij})\in\alpha_{ij}^{m_{ij}}S\), \(0\leq i<j\leq l\}\) of multi-derivations is a free \(S\)-module. In this case \(\mathfrak m\) is called a free multiplicity.
Let \(\Lambda_l^b\) denote the cone of multiplicities in \(\Lambda_l\) satisfying the inequalities \(m_{ij}\leq m_{ik}+m_{jk}+1\) for every triple \(i, j, k\). (\(\Lambda_l^b\) is called the balanced cone of multiplicities.) In [T. Abe et al., J. Lond. Math. Soc., II. Ser. 80, No. 1, 121–134 (2009; Zbl 1177.32017)], the authors completely characterized free multiplicities \(\mathfrak m\in\Lambda_l^b\) of the form \(m_{ij} = n_i+n_j+\epsilon_{ij}\) where \(n_0,\dots,n_l\in\mathbb Z_{\geq 0}\) and \(\epsilon_{ij}\in\{-1, 0, 1\}\) \((0\leq i < j\leq l)\). Such a multiplicity is called an ANN multiplicity.
The main result of the paper under review states that the free ANN multiplicities are the only free multiplicities in \(\Lambda_l^b\). More precisely, let \(K_{l+1}\) be a complete graph on \(l+1\) vertices \(\{v_0,\dots, v_l\}\) naturally associated with \(A_l\) (the hyperplane \(H_{ij}\in A_l\) corresponds to the edge \(\{v_i, v_j\}\) in \(K_{l+1}\)). Then a multiplicity \(\mathfrak m\) on \(A_{l}\) yields a labeling of the edges of \(K_{l+1}\) by assigning \(m_{ij}\) to an edge \(\{v_i, v_j\}\). A three-cycle \(\{v_i, v_j\}\), \(\{v_j, v_k\}\), \(\{v_k,v_i\}\) in \(K_{l+1}\) is said to be an odd three-cycle of \(\mathfrak m\) if the integer \(m_{ij}+m_{jk}+m_{ki}\) is odd. If \(C\) is a four-cycle with edges \(\{v_i, v_j\}\), \(\{v_j,v_s\}\), \(\{v_s,v_t\}\), \(\{v_t, v_i\}\) in \(K_{l+1}\), then one sets \(\mathfrak m(C)= |m_{ij}-m_{js}+m_{st}-m_{ti}|\) (this number is well-defined; it does not depend on the ordering of the edges of \(C\)).
If \(U\) is a set of at least four vertices of \(K_{l+1}\), then \(\mathfrak m_U\) denotes the restriction of \(\mathfrak m\) to the subset \(\{H_{ij}\mid\{v_i,v_j\}\subset U\}\) and the deviation of \(\mathfrak m\) over \(U\) is defined by \(\mathrm{DV}(\mathfrak m_U) = \sum_{C\subset U}\mathfrak m(C)^2\) (\(C\) runs over all four-cycles of \(K_{l+1}\) contained in \(U\)). Furthermore, let \(q_U\) denote the number of odd three-cycles of \(\mathfrak m\) contained in \(U\).
With this notation, the main result of the paper is the following theorem. Let \((A_l,\mathfrak m)\) be a multi-braid arrangement with \(\mathfrak m\in\Lambda_l^b\). Then the following conditions are equivalent; (1) \((A_l,\mathfrak m)\) is free; (2) \(\mathrm{DV}(\mathfrak m_U)\leq q_U(|U|-1)\) for every set \(U\subset\{v_0,\dots, v_l\}\) with at least four vertices; (3) \(\mathfrak m\) is a free ANN multiplicity.
The paper also contains some illustrating examples and a conjecture about the structure of all free multiplicities on braid arrangements.


13N15 Derivations and commutative rings
05E40 Combinatorial aspects of commutative algebra
14N20 Configurations and arrangements of linear subspaces


Zbl 1177.32017


Full Text: DOI arXiv


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