Homological properties of determinantal arrangements (Q342840)

In this really interesting article the author studies determinantal arrangements in the context of the freeness and the topology of their complements.NEWLINENEWLINELet us recall some notions before we present the main results of the article. Let \(X=\mathbb C^n\) with the coordinate ring \(R=\mathbb C[x_{1},\dots,x_n]\). We denote by \(\mathrm{Der}_X\) the free \(R\)-module of vector fields on \(X\) generated by \(\{\partial/\partial_{x_i}\}_{i=1}^n\). For a reduced element \(f\in R\) and divisor \(D=\mathrm{Var}(f)\), we can define the module of logarithmic derivations along \(D\) as NEWLINE\[NEWLINE\mathrm{Der}_X(-\log (D))=\{\theta\in\mathrm{Der}_{X}:\theta(f)\in (f)\}.NEWLINE\]NEWLINE We say that a divisor \(D\) on \(X\) is free if \(\mathrm{Der}_X(-\log(D))\) is a free \(R\)-module.NEWLINENEWLINELet \(M\) be the \(2\times n\) matrix of indeterminates NEWLINE\[NEWLINEM=\left(\begin{aligned} & x_1 x_{2}\dots x_{n} \\ & y_1 y_2\dots y_n\end{aligned}\right).NEWLINE\]NEWLINE For \(i<j\), let \(\Delta_{ij}\) denote \(2\)-minor of \(M\) using the \(i\)-th and \(j\)-th columns, \(\Delta_{ij} = x_iy_j-x_jy_i\).NEWLINENEWLINEIt is known that for each graph \(G\) one can associate a hyperplane arrangement. The same procedure can be performed using minors of \(M\). \textit{A determinantal arrangement} \(\mathcal A_G\) is defined as the arrangement of the determinantal varieties \(\mathrm{Var}(\Delta_{ij})\) for each edge between vertices \(v_i\) and \(v_j\) of \(G\). At last, we say that a graph \(G\) is \textit{chordal} if and only if there exists an ordering of vertices, such that for each vertex \(v\), the induced subgraph on \(v\) and its neighbors that occur before it in the sequence ais an complete graph.NEWLINENEWLINENow we are ready to present main results.NEWLINENEWLINETheorem 1. Let \(G\) be the complete graph on \(n\) vertices for \(n \geq 3\). The determinantal arrangement \(\mathcal A_G\) is free.NEWLINENEWLINETheorem 2. If a determinantal arrangement \(\mathcal A_G\) is free, then \(G\) is chordal. Moreover, if \(G\) has a chord-free induced cycle of length \(k\), then NEWLINE\[NEWLINE\mathrm{pdim}_R(\mathrm{Der}_X(-\log(\mathcal A_G))\geq k-3,NEWLINE\]NEWLINE where \(\mathrm{pdim}\) denotes the projective dimension.NEWLINENEWLINEThe second part of the paper is devoted to the topology of the complements of free determinantal arrangements. The main result of this part is devoted to Poincaré polynomials.NEWLINENEWLINETheorem 3. Let \(G\) be the complete graph on \(n\) vertices for \(n\geq 2\). Let \(U_n=\mathbb C^{2n}\setminus\mathcal A_G\), then NEWLINE\[NEWLINE\mathrm{Poin}(U_n,t)=(1+t^3)(1+t)^{n-1} \prod_{k=1}^{n-2}(1+kt).NEWLINE\]NEWLINE Theorem 4. Let \(G\) be a chorodal graph, then Poincaré polynomial of \(U_n=\mathbb C^{2n}\setminus\mathcal A_G\) factors over \(\mathbb Q\) into a product of a cubic with \(2|\mathcal A_G|- 3\) linear terms.NEWLINENEWLINEThe article is nicely written and provided proofs are rather elementary.