Gin and lex of certain monomial ideals - Murai, Satoshi; Hibi, Takayuki (MATHEMATICA SCANDINAVICA, 2006)
Let A = K[x_1,…, x_n] denote the polynomial ring in n variables over a field K of characteristic 0 with each x_i = 1. Given arbitrary integers i and j with 2 ≤ i ≤ n and 3 ≤ j ≤ n, we will construct a monomial ideal I A such that (i) β_k(I) < β_k(Gin(I)) for all k < i, (ii) β_i(I)= β_i(Gin(I)), (iii) β_((Gin(I)) < β_((Lex(I)) for all < j and (iv) β_j(Gin(I)) = β_j(Lex(I)), where Gin(I) is the generic initial ideal of I with respect to the reverse lexicographic order induced by x_1 > … > x_n and where Lex(I) is the lexsegment ideal with the same Hilbert function as I.