On the dual graph of Cohen-Macaulay algebras

TL;DR

This paper explores the properties of the dual graph of Cohen-Macaulay algebras, providing quantitative bounds on connectivity and diameter, extending classical theorems with algebraic and combinatorial insights.

Contribution

It introduces two new bounds relating the dual graph's connectivity and diameter to algebraic invariants and geometric configurations, extending Hartshorne's connectedness theorem.

Findings

Gorenstein subspace arrangements have dual graphs that are r-connected, where r is the Castelnuovo-Mumford regularity.

For arrangements of lines with no three meeting, the dual graph's diameter is bounded by the codimension.

Bounds are sharp and extend classical combinatorial theorems to algebraic settings.

Abstract

Given a projective algebraic set X, its dual graph G(X) is the graph whose vertices are the irreducible components of X and whose edges connect components that intersect in codimension one. Hartshorne's connectedness theorem says that if (the coordinate ring of) X is Cohen-Macaulay, then G(X) is connected. We present two quantitative variants of Hartshorne's result: 1) If X is a Gorenstein subspace arrangement, then G(X) is r-connected, where r is the Castelnuovo-Mumford regularity of X. (The bound is best possible; for coordinate arrangements, it yields an algebraic extension of Balinski's theorem for simplicial polytopes.) 2) If X is a canonically embedded arrangement of lines no three of which meet in the same point, then the diameter of the graph G(X) is not larger than the codimension of X. (The bound is sharp; for coordinate arrangements, it yields an algebraic expansion on the recent combinatorial result that the Hirsch conjecture holds for flag normal simplicial complexes.)