Powers of generalized binomial edge ideals of path graphs (Q6593813)

In this paper, the authors study powers of the generalized binomial edge ideal \(J_{{K_m},P_n}\) of a path graph \(P_n\), where \(K_m\) is the complete graph with \(m\) vertices. They explicitly compute the regularities and determine the limit of their depths. They also show that these ordinary powers coincide with their symbolic powers. Additionally, they study the Rees algebra and the special fiber ring of \(J_{{K_m},P_n}\) utilizing Sagbi basis theory. In particular, the authors obtain precise formulas for the regularity of these blowup algebras.