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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.

MSC:

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

Citations:

Zbl 1177.32017

Software:

Macaulay2
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References:

[1] T. Abe, K. Nuida and Y. Numata, Signed-eliminable graphs and free multiplicities on the braid arrangement, arXiv:0712.4110v3. · Zbl 1177.32017 · doi:10.1112/jlms/jdp019
[2] T. Abe, K. Nuida and Y. Numata, Signed-eliminable graphs and free multiplicities on the braid arrangement, J. Lond. Math. Soc. (2) 80 (2009), no. 1, 121-134. · Zbl 1177.32017 · doi:10.1112/jlms/jdp019
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