Gin AND Lex OF CERTEIN MONOMIAL IDEALS

Let A = K[x1, . . . , xn] denote the polynomial ring in n variables over a field K of characteristic 0 with each deg xi = 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) βk(I) < βk(Gin(I)) for all k < i, (ii) βi(I) = βi(Gin(I)), (iii) βl(Gin(I)) < βl(Lex(I)) for all l < j and (iv) βj(Gin(I)) = βj(Lex(I)), where Gin(I) is the generic initial ideal of I with respect to the reverse lexicographic order induced by x1 > · · · > xn and where Lex(I) is the lexsegment ideal with the same Hilbert function as I.