Gin and lex of certain monomial ideals

Authors

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

$I$ .