Powers of generalized binomial edge ideals of path graphs

TL;DR

This paper investigates algebraic properties of powers of generalized binomial edge ideals associated with path graphs, providing explicit formulas for regularity, depth limits, and algebraic structures like Rees algebras.

Contribution

It offers explicit computations of regularity, depth limits, and analyzes the Rees algebra and fiber ring of these ideals, advancing understanding of their algebraic behavior.

Findings

Explicit formulas for regularity of powers

Limit of depths of the ideals

Rees algebra and fiber ring structures analyzed

Abstract

In this article, we study the powers of the generalized binomial edge ideal $\mathcal{J}_{K_m,P_n}$ of a path graph $P_n$. We explicitly compute their regularities and determine the limit of their depths. We also show that these ordinary powers coincide with their symbolic powers. Additionally, we study the Rees algebra and the special fiber ring of $\mathcal{J}_{K_m,P_n}$ via Sagbi basis theory. In particular, we obtain exact formulas for the regularity of these blowup algebras.