Let Hilbp be the Hilbert scheme parametrizing the closed subschemes of with Hilbert polynomial over a field K of characteristic zero. By bounding below the cohomological Hilbert functions of the points of Hilbp we define locally closed subspaces of the Hilbert scheme. The aim of this paper is to show that some of these subspaces are connected. For this we exploit the edge ideals constructed by D. Mall in [D. Mall, Connectedness of Hilbert function strata and other connectedness results, J. Pure Appl. Algebra 150 (2000) 175–205]. It turns out that these ideals are sequentially Cohen–Macaulay and that their initial ideals with respect to the reverse lexicographic term order are generic initial ideals.