Nonunimodal Gorenstein sequences of higher socle degrees

We study Gorenstein h-vectors (h(0), h(1)...,h(e)) of socle degree e with h(1) >= h(i) for each i, and find a necessary and sufficient condition that there exists a nonunimodal Gorenstein sequence in terms of socle degree e and codimension h(1). In particular, we prove that there exist nonunimodal Gorenstein h-vectors if and only if h(1) >= 4e - 3 for e >= 4. We also find infinitely many cases of non-Gorenstein h -vectors having the lower bound in [15]. This result generalizes the recent work [17] that the h -vector (1, 12, 11,12,1) is not a Gorenstein sequence. 2017 Elsevier Inc. All rights reserved.