Regularity of Line Configurations

Bruno Benedetti; Michela Di Marca; Matteo Varbaro

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arXiv · 1608.02134

Regularity of Line Configurations

Bruno Benedetti·Michela Di Marca·Matteo Varbaro

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Abstract

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.

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Bruno Benedetti, Michela Di Marca, Matteo Varbaro. 2016-08-06. Regularity of Line Configurations. https://arxiv.org/abs/1608.02134

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