Symbolic Rees algebras of complementary edge ideals

Abstract

Let $G$ be a finite simple graph on $[n]$ and let $I_c(G)$ denote its complementary edge ideal in the polynomial ring $S=K[x_1,\dots ,x_n]$. We give a combinatorial description, in terms of the structure of $G$, of the minimal generators of the symbolic Rees algebra ${\mathcal{R}}_s(I_c(G))={⨁}_{k\ge 0}{I}_c(G{)}^{(k)}{t}^k$, and show that this algebra is generated in degree at most $6$. Moreover, we completely determine the minimal generators of ${\mathcal{R}}_s(I_c(G))$ in graph-theoretic terms. We then study in more detail the homological invariants of the symbolic powers $I_c(G{)}^{(k)}$ for the classes of cycle graphs and complete multipartite graphs. For theses families, we study the behavior of the symbolic depth function $k\mapsto \mathrm{depth}S/I_c(G{)}^{(k)}$, we obtain the limit depth of the symbolic powers and the Waldschmidt constant of $I_c(G)$, and further prove that all the symbolic powers $I_c(G{)}^{(k)}$ are componentwise linear.

33 pages