Depth of an initial ideal

Abstract

Given an arbitrary integer $d>0$, we construct a homogeneous ideal $I$ of the polynomial ring $S=K[x_1,\dots ,x_{3d}]$ in $3d$ variables over a filed $K$ for which $S/I$ is a Cohen--Macaulay ring of dimension $d$ with the property that, for each of the integers $0\le r\le d$, there exists a monomial order ${<}_r$ on $S$ with $\mathrm{depth}(S/{\mathrm{in}}_{{<}_r}(I))=r$, where ${\mathrm{in}}_{{<}_r}(I)$ is the initial ideal of $I$ with respect to ${<}_r$.

4 pages