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

We find a sufficient condition that image is not level based on a reduction number. In particular, we prove that a graded Artinian algebra of codimension 3 with Hilbert function image cannot be level if hd≤2d+3, and that there exists a level O-sequence of codimension 3 of type image for hd≥2d+k for k≥4. Furthermore, we show that image is not level if image, and also prove that any codimension 3 Artinian graded algebra A=R/I cannot be level if image. In this case, the Hilbert function of A does not have to satisfy the condition hd−1>hd=hd+1. Moreover, we show that every codimension n 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) for every d>θ where h0cdots, three dots, centered>hs−1>hs. In particular, we show that if A is of codimension 3, then (hd−1−hd)<2(hd−hd+1) for every θ