On the Diameter of Dual Graphs of Stanley-Reisner Rings and Hirsch Type Bounds on Abstractions of Polytopes

Abstract

Let R be an equidimensional commutative Noetherian ring of positive dimension. The dual graph of R is defined as follows: the vertices are the minimal prime ideals of R, and the edges are the pairs of prime ideals (P1, P2) with height (P1 + P2) = 1. If R satisfies Serre's property (S2), then G(R) is connected. In this note, we provide lower and upper bounds for the maximum diameter of dual graphs of Stanley-Reisner rings satisfying (S2). These bounds depend on the number of variables and the dimension. Dual graphs of (S2) Stanley-Reisner rings are a natural abstraction of the 1-skeletons of polyhedra. We discuss how our bounds imply new Hirsch-type bounds on 1-skeletons of polyhedra.

Square-free Gröbner degenerations

“…If I ∆ is quadratic (that is ∆ is flag), then I ∆ is Hirsch by [AB14]. If height(I ∆ ) ≤ 3, then it is simple to check that I ∆ is Hirsch (see [Ho16,Corollary A.4] for the less trivial case in which height(I ∆ ) = 3). In each case, we conclude because, under the assumptions of the theorem, by [DV17, Theorem 3.3] the diameter of the dual graph of I is bounded above from that of in(I).…”

A generalized Serre’s condition

“…In Section 4, we consider the Hochster-Huneke graph of R (denoted by G(R)) where R is a local ring or a quotient of a polynomial ring and a homogeneous ideal. It is known that a Stanley-Reisner ring satisfies (S 2 ) if and only if every localization of R at a prime has a connected Hochster-Huneke graph [Kum08,Hol18]. We create a generalization of the Hochster-Huneke graph.…”