GORENSTEIN SEQUENCES OF HIGH SOCLE DEGREES

In [4], the authors showed that if an h-vector (h0, h1, ..., he) with h1 = 4e - 4 and hi ≤ h1 is a Gorenstein sequence, then h1 = hi for every 1 ≤ i ≤ e - 1 and e ≥ 6. In this paper, we show that if an h-vector (h0, h1, ..., he) with h1 = 4e - 4, h2 = 4e - 3, and hi ≤ h2 is a Gorenstein sequence, then h2 = hi for every 2 ≤ i ≤ e - 2 and e ≥ 7. We also propose an open question that if an h-vector (h0, h1, ..., he) with h1 = 4e - 4, 4e - 3 < h2 ≤ (h1)(1)|+1+1, and h2 ≤ hi is a Gorenstein sequence, then h2 = hi for every 2 ≤ i ≤ e - 2 and e ≥ 6.