Depth of initial ideals of normal edge rings (1101.4058v1)

Abstract: Let $G$ be a finite graph on the vertex set $[d] = {1, ..., d }$ with the edges $e_1, ..., e_n$ and $K[\tb] = K[t_1, ..., t_d]$ the polynomial ring in $d$ variables over a field $K$. The edge ring of $G$ is the semigroup ring $K[G]$ which is generated by those monomials $\tb^e = t_it_j$ such that $e = {i, j}$ is an edge of $G$. Let $K[\xb] = K[x_1, ..., x_n]$ be the polynomial ring in $n$ variables over $K$ and define the surjective homomorphism $\pi : K[\xb] \to K[G]$ by setting $\pi(x_i) = \tb^{e_i}$ for $i = 1, ..., n$. The toric ideal $I_G$ of $G$ is the kernel of $\pi$. It will be proved that, given integers $f$ and $d$ with $6 \leq f \leq d$, there exist a finite connected nonbipartite graph $G$ on $[d]$ together with a reverse lexicographic order $<{\rev}$ on $K[\xb]$ and a lexicographic order $<{\lex}$ on $K[\xb]$ such that (i) $K[G]$ is normal, (ii) $\depth K[\xb]/\ini_{<{\rev}}(I_G) = f$ and (iii) $K[\xb]/\ini{<{\lex}}(I_G)$ is Cohen--Macaulay, where $\ini{<{\rev}}(I_G)$ (resp.\ $\ini{<{\lex}}(I_G)$) is the initial ideal of $I_G$ with respect to $<{\rev}$ (resp.\ $<{\lex}$) and where $\depth K[\xb]/\ini{<{\rev}}(I_G)$ is the depth of $K[\xb]/\ini{<_{\rev}}(I_G)$.