Regularity of powers of cover ideals of unimodular hypergraphs (Q1786458)

Let \(n\) be a positive integer, let \(\mathcal{V}\) = \([n]:=\{1,2, \dots, n\}\), and let \(\mathcal{E}\) be a set of nonempty subsets of \(\mathcal{V}\). The pair \(\mathcal{H}=(\mathcal{V}, \mathcal{E})\) is called a hypergraph with vertex set \(\mathcal{V}\) and edge set \(\mathcal{E}\). The rank of \(\mathcal H\), denoted by \(\mathrm{rank}(\mathcal{H})\), is the maximum number of any of the edges in \(\mathcal{H}\). A hypergraph \(\mathcal{H}\) can be also defined by its incidence matrix \(A(\mathcal{H})\), and \(\mathcal{H}\) is said to be unimodular in case its incidence matrix has the property that the determinant of every square submatrix of \(A(\mathcal{H})\) is \(0\), \(1\), or \(-1\). If \(J(\mathcal H)\) is the cover ideal of \(\mathcal{H}\) in the polynomial ring \(R=K[X_1, \dots, X_n]\) over a field \(K\), then the authors prove, among others, that reg \(J({\mathcal H})^s\) is a linear function in \(s\) for all \(s\geqslant r[n/2] +1\), where \(r=\mathrm{rank} (\mathcal{H})\).