On the Dual Graphs of Cohen–Macaulay Algebras

Given an equidimensional algebraic set |X⊂ P ⁿ|⁠, 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 an arrangement of lines no three of which meet in the same point, and |X| is canonically embedded in |P ⁿ|⁠, then the diameter of the graph |$G(X)$| is less than or equal to |codim P ⁿX|⁠. (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.) On the way to these results, we show that there exists a graph which is not the dual graph of any simplicial complex (no matter the dimension).