Gin and lex of certain monomial ideals

Abstract

Let $A = K[x_1,\ldots, x_n]$ denote the polynomial ring in $n$ variables over a field $K$ of characteristic $0$ with each $\deg x_i = 1$. Given arbitrary integers $i$ and $j$ with $2 \leq i \leq n$ and $3 \leq j \leq n$, we will construct a monomial ideal $I \subset A$ such that (i) $\beta_k(I) < \beta_k(\mathrm{Gin}(I))$ for all $k < i$, (ii) $\beta_i(I)= \beta_i(\mathrm{Gin}(I))$, (iii) $\beta_\ell((\mathrm{Gin}(I)) < \beta_\ell((\mathrm{Lex}(I))$ for all $\ell < j$ and (iv) $\beta_j(\mathrm{Gin}(I)) = \beta_j(\mathrm{Lex}(I))$, where $\mathrm{Gin}(I)$ is the generic initial ideal of $I$ with respect to the reverse lexicographic order induced by $x_1 > \cdots > x_n$ and where $\mathrm{Lex}(I)$ is the lexsegment ideal with the same Hilbert function as $I$.