Regularity of Line Configurations

TL;DR

This paper explores the algebraic and combinatorial properties of line arrangements and simplicial complexes, revealing their regularity and nerve complex representations, bridging geometry, algebra, and topology.

Contribution

It establishes that arithmetically-Gorenstein line arrangements have uniform intersection numbers linked to Castelnuovo-Mumford regularity and characterizes certain simplicial complexes as nerve complexes of graded algebras.

Findings

Line arrangements have uniform intersection numbers equal to their algebraic regularity.

Certain simplicial complexes can be realized as nerve complexes of graded algebras.

Provides a converse to a recent algebraic-topological correspondence.

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.