Asymptotic invariants of symbolic powers of binomial edge ideals

Abstract

To a graph $G$ one associates the binomial edge ideal $J_G$ generated by a collection of binomials corresponding to the edges of $G$. In this paper, we study the asymptotic behavior of symbolic powers of $J_G$, its lexicographic initial ideal ${\mathrm{in}}_{<}(J_G)$, and its multigraded generic initial ideal $\mathrm{gin}(J_G)$. We focus on the Waldschmidt constant, $\hat{\alpha }$, and asymptotic regularity, $\widehat{\mathrm{reg}}$, which capture linear growth of minimal generator degrees and Castelnuovo--Mumford regularity. We explicitly compute $\hat{\alpha }(J_G)$ and $\hat{\alpha }({\mathrm{in}}_{<}(J_G))$, and compare the Betti numbers of the symbolic powers of $J_G$ and $J_H$, where $H$ is a subgraph of $G$. To analyze ${\mathrm{in}}_{<}(J_G)$ and $\mathrm{gin}(J_G)$, we use the symbolic polyhedron, a convex polyhedron that encodes the elements of the symbolic powers of a monomial ideal. We determine its vertices via $G$'s induced connected subgraphs and show that $\hat{\alpha }(\mathrm{gin}(J_G))=\hat{\alpha }(I_G)$, where $I_G$ is the edge ideal of $G$. This yields an alternate proof of known bounds for $\hat{\alpha }(I_G)$ in terms of $G$'s clique number and chromatic number.

25 pages. Comments welcome