Koszul multi-Rees algebras of principal \(L\)-Borel ideals (Q2029203)

Let \(R=k[x_1,\ldots,x_n]\), where \(k\) is a field, and \(L\) be a linear ordering on a subset of the variables of \(R\). Given a monomial \(m \in R\), a variable \(x_j\) dividing \(m\) and any \(x_i\) such that \(x_i >_L x_j\), the monomial \(\frac{x_i}{x_j} m\) is called an \(L\)-Borel move on \(m\). A monomial ideal \(I \subseteq R\) is called \(L\)-Borel if, given a monomial \(m \in I\), any \(L\)-Borel move on \(m\) still belongs to \(I\). Finally, \(I\) is called principal \(L\)-Borel if every minimal monomial generator of \(I\) can be obtained by \(L\)-Borel moves on a single monomial. In the article under review, the authors consider \(\mathcal I=\{I_1,\ldots,I_r\}\) principal \(L_i\)-Borel ideals (the linear posets \(L_i\) can be different), and they consider the multi-Rees algebra \(R[\mathcal I \mathbf{t}] = \bigoplus_{a_1,\ldots,a_r \geq 0} I_1^{a_1} \cdots I_r^{a_r}t_1^{a_1} \cdots t_r^{a_r}\). Starting from \(\mathcal I\), the authors construct a bipartite incidence graph \(G(\mathcal I)\). The main result states that if \(G(\mathcal I)\) is chordal bipartite, then \(R[\mathcal I \mathbf{t}]\) is G-quadratic, and hence Koszul. This result is a significant improvement of a theorem of \textit{H. Ohsugi} and \textit{T. Hibi}, stating that the toric ring of the edge ideal of a graph is Koszul if the graph is chordal bipartite [Adv. Appl. Math. 22, No. 1, 25--28 (1999; Zbl 0916.05046)].