Regularity of line configurations

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 ≥3. This provides the converse of a recent result by Katzman, Lyubeznik and Zhang.