Generic initial ideals and graded Artinian-level algebras not having the Weak-Lefschetz Property

We find a sufficient condition that H is not level based on a reduction number. In particular, we prove that a graded Artinian algebra of codimension 3 with Hilbert function H=(h0,h1,…,hd−1>hd=hd+1) cannot be level if hd≤2d+3hd≤2d+3, and that there exists a level O-sequence of codimension 3 of type H for hd≥2d+khd≥2d+k for k≥4k≥4. Furthermore, we show that H is not level if β1,d+2(Ilex)=β2,d+2(Ilex), and also prove that any codimension 3 Artinian graded algebra A=R/IA=R/I cannot be level if β1,d+2(Gin(I))=β2,d+2(Gin(I)). In this case, the Hilbert function of AA does not have to satisfy the condition hd−1>hd=hd+1hd−1>hd=hd+1.Moreover, we show that every codimension nn graded Artinian level algebra having the Weak-Lefschetz Property has a strictly unimodal Hilbert function having a growth condition on (hd−1−hd)≤(n−1)(hd−hd+1)(hd−1−hd)≤(n−1)(hd−hd+1) for every d>θd>θ where h0⋯>hs−1>hs.h0⋯>hs−1>hs. In particular, we show that if AA is of codimension 3, then (hd−1−hd)<2(hd−hd+1)(hd−1−hd)<2(hd−hd+1) for every θ0andsoc(A)d−1=0 for some r1(A)