  We show that in arithmetically-Gorenstein line arrangements with only planar
singularities, each line intersects the same number of other lines. This number
has an algebraic interpretation: it is the Castelnuovo-Mumford regularity of
the coordinate ring of the arrangement.
  We also prove that every (d-1)-dimensional simplicial complex whose 0-th and
1-st homologies are trivial is the nerve complex of a suitable d-dimensional
standard graded algebra of depth $\ge 3$. This provides the converse of a
recent result by Katzman, Lyubeznik and Zhang.
