Heavy hyperplanes in multiarrangements and their freeness (Q1631393)

The authors introduce and study the so-called unbalanced multiarrangements in the context of the freeness. In order to formulate the main results, we need to recall basics on multiarrangements. Let \(V\) be a vector space of dimension \(\ell\) over a field \(\mathbb{K}\) and \(S = S^{*}(V)\) be the symmetric algebra. A central arrangement of hyperplanes \(\mathcal{A}\) is a finite collection of (linear) hyperplanes. By a multiarrangement we mean a pair \((\mathcal{A},m)\), where \(\mathcal{A}\) is an arrangement of hyperplanes and \(m : \mathcal{A} \rightarrow \mathbb{Z}_{>0}\) is a multiplicity function. Define \(|m| = \sum_{H \in \mathcal{A}} m(H)\). Denote by \(L(\mathcal{A})\) the intersection lattice of \(\mathcal{A}\). For any \(X \subset V\), let \(\mathcal{A}_{X} = \{ H \in \mathcal{A} : X \subseteq H\}\) with \(m_{X} = m|_{\mathcal{A}_{X}}\) be the localization of a multiarrangement \((\mathcal{A},m)\) at \(X\). Moreover, we define \(\mathcal{A}^{H} = \{ H \cap H' : H' \in \mathcal{A} \setminus H \}\) and \(m^{H}(X) := | \mathcal{A}_{X} \setminus \{H\}|\) for \(X \in\mathcal{A}^{H}\). For \(p\geq 1\), the \(S\)-module \(\text{Der}^{p}(S)\) is the set of all alternating \(p\)-linear forms \(\theta : S^{p} \rightarrow S\) such that \(\theta\) is a \(\mathbb{K}\)-derivation in each variable. For \(p=0\) one defines \(\text{Der}^{0}(S) := S\). The logarithmic derivation modules of \((\mathcal{A},m)\) are defined as \[ D^{p}(\mathcal{A},m):= \bigg\{ \theta \in \text{Der}^{p}(S) : \theta(\alpha_{H},f_{2}, \dots, f_{p}) \in \alpha_{H}^{m(H)}S \text{ for all } H \in \mathcal{A} \text{ and } f_{2}, \dots,f_{p}\in S \bigg\}, \] where \(\alpha_{H}\) denotes the defining linear equation of \(H\). If \(D^{1}(\mathcal{A},m) := D(\mathcal{A},m)\) is a free \(S\)-module, we call \((\mathcal{A},m)\) a free multiarrangement. Now for a multiarrangement \((\mathcal{A},m)\) we define a function in \(x\) and \(t\): \[ \psi(\mathcal{A},m;x,t) = \sum_{p=0}^{\ell} H(D^{p}(\mathcal{A},m),x)(t(x-1)-1)^{p}, \] where \(H(D^{p}(\mathcal{A},m),x)\) is the Hilbert series of the graded \(S\)-module \(D^{p}(\mathcal{A},m)\). A characteristic polynomial of a multiarrangement \((\mathcal{A},m)\) is defined as \[ \chi(\mathcal{A},m;t) = \lim_{x\rightarrow 1} (-1)^{\ell} \psi(\mathcal{A},m;x,t). \] We define the Betti number \(b_{i}(\mathcal{A},m)\) of \((\mathcal{A},m)\) by \[ \chi(\mathcal{A},m;t) = t^{\ell} - b_{1}(\mathcal{A},m)t^{\ell -1} + b_{2}(\mathcal{A},m)t^{\ell -2} -\dots + (-1)^{\ell}b_{\ell}(\mathcal{A},m). \] Now we are ready to present the main definition of the paper. Definition. Let \((\mathcal{A},m)\) be a multiarrangement in \(V = \mathbb{K}^{\ell}\). {\parindent=6mm \begin{itemize}\item[1)] We say that a hyperplane \(H_{0} \in \mathcal{A}\) is a heavy hyperplane with \(m_{0} = m(H_{0})\) if \(2m_{0} \geq |m|\). A multiarrangement \((\mathcal{A},m)\) is called unbalanced if it has a heavy hyperplane \(H_{0} \in \mathcal{A}\). \item[2)] We say that \((\mathcal{A},m)\) has a locally heavy hyperplane \(H_{0} \in \mathcal{A}\) with \(m_{0}:= m(H_{0})\) if \(2m_{0} \geq |m_{X}|\) for all localization \((\mathcal{A}_{X},m_{X})\) with \(X \in \mathcal{A}^{H_{0}}\) satisfying \(|\mathcal{A}_{X}| \geq 3\). \item[3)] Let \(H_{0}\) be a locally heavy hyperplane in \((\mathcal{A},m)\). The Euler-Ziegler restriction \((\mathcal{A}^{H_{0}},m^{H_{0}})\) of \((\mathcal{A},m)\) onto \(H_{0} \in \mathcal{A}\) is defined by \(\mathcal{A}^{H_{0}} := \{ H_{0} \cap H : H \in \mathcal{A} \setminus \{H_{0}\}\}\) and \(m^{H_{0}}(X) = \sum_{H \in \mathcal{A}_{X} \setminus \{H_{0}\}}m(H)\). \item[4)] Let \((\mathcal{A},m)\) be an arbitrary multiarrangement and \(H_{0} \in \mathcal{A}\). The multi-Ziegler restriction \((\mathcal{A}^{H_{0}}, m^{H_{0}})\) of \((\mathcal{A},m)\) onto \(H_{0}\) is defined by \(\mathcal{A}^{H_{0}}:= \{H_{0} \cap H : H \in \mathcal{A} \setminus \{H_{0}\}\}\) and \(m^{H_{0}}(X) : = \sum_{H \in \mathcal{A}_{X} \setminus \{ H_{0} \}} m(H)\). \end{itemize}} Now we are ready to formulate the main results of the paper. Theorem A. {\parindent=6mm \begin{itemize}\item[1)] Let \((\mathcal{A},m)\) be unbalanced with a heavy hyperplane \(H_{0} \in \mathcal{A}\) with \(m_{0} = m(H_{0})\). Then \[ b_{2}(\mathcal{A},m) - m_{0}(|m|-m_{0}) \geq b_{2}(\mathcal{A}^{H_{0}},m^{H_{0}}). \] Moreover, \((\mathcal{A},m)\) is free iff the Euler-Ziegler restriction \((A^{H_{0}},m^{H_{0}})\) of \((\mathcal{A},m)\) onto \(H_{0}\) is free and \(b_{2}(\mathcal{A},m) - m_{0}(|m| - m_{0}) = b_{2}(\mathcal{A}^{H_{0}},m^{H_{0}})\). \item[2)] Let \((\mathcal{A},m)\) have a locally heavy hyperplane \(H_{0} \in \mathcal{A}\) with \(m_{0} = m(H_{0})\). Then \((\mathcal{A},m)\) is free if the Euler-Ziegler restriction \((\mathcal{A}^{H_{0}},m^{H_{0}})\) of \((\mathcal{A},m)\) onto \(H_{0}\) is free and \(b_{2}(\mathcal{A},m) - m_{0}(|m|-m_{0}) = b_{2}(\mathcal{A}^{H_{0}},m^{H_{0}})\). \end{itemize}} Theorem B. Let \((\mathcal{A},m)\) be a multiarrangement and \(H\in \mathcal{A}\). Then \[ b_{2}^{H}(\mathcal{A},m) \geq B_{2}^{H}(\mathcal{A},m) \geq b_{2}(\mathcal{A}^{H},m^{H}) \geq b_{2}(\mathcal{A}^{H},m^{*}), \] where \(m^{*}\) is the Euler multiplicity of \((\mathcal{A},m)\) onto \(H\), \((\mathcal{A}^{H},m^{H})\) is the multi-Ziegler restriction, \[ b_{2}^{H}(\mathcal{A},m) = b_{2}(\mathcal{A},m) - m(H)(|m| - m(H)), \] and \[ B_{2}^{H}(\mathcal{A},m) = \sum_{X \in L_{2}(\mathcal{A}) \setminus \mathcal{A}^{H}} b_{2}(\mathcal{A}_{X},m_{X}). \] Theorem C. Let \(\mathcal{A}\) be an arrangement in \(V\) and let \(\{X_{i}\}_{i=0}^{\ell}\) be a flag of \(\mathcal{A}\) such that \(X_{i+1} \in \mathcal{A}^{X_{i}}\) is heavy in \((\mathcal{A}^{X_{i}},m^{X_{i}})\) for \(i=0,\dots, \ell -1\). Then \(\mathcal{A}\) is free if and only if \[ b_{2}(\mathcal{A}) = \sum_{0 \leq i < j \leq \ell -1} m^{X_{i}}(X_{i+1})m^{X_{j}}(X_{j+1}). \]