Gin and Lex of certain monomial ideals

Original · EN

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