Nonunimodal Gorenstein sequences of higher socle degrees

We study Gorenstein h-vectors (h0,h1,…,he) of socle degree e with h1≥hi 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 h1. In particular, we prove that there exist nonunimodal Gorenstein h-vectors if and only if h1≥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.