Regulating Hartshorne’s connectedness theorem

Abstract

A classical theorem by Hartshorne states that the dual graph of any arithmetically CohenMacaulay projective scheme is connected. We give a quantitative version of Hartshorne's result, in terms of Castelnuovo-Mumford regularity. If X ⊂ P n is an arithmetically Gorenstein projective scheme of regularity r + 1, and if every irreducible component of X has regularity ≤ r , we show that the dual graph of X is r+r −1 r -connected. The bound is sharp. We also provide a strong converse to Hartshorne's result: Every connected graph is the dual graph of a suitable arithmetically Cohen-Macaulay projective curve of regularity ≤ 3, whose components are all rational normal curves. The regularity bound is smallest possible in general.Further consequences of our work are: (1) Any graph is the Hochster-Huneke graph of a complete equidimensional local ring. (This answers a question by Sather-Wagstaff and Spiroff.) (2) The regularity of a curve is not larger than the sum of the regularities of its primary components.

On the diameter of an ideal

“…Proof. Following the proofs of [BBV,Theorem 4.3 and Corollary 4.4], the removal of a set of vertices such that the sum of the degrees of the corresponding minimal primes is at most r − 1 does not disconnect G(I). Therefore G(I) is (r, w)-connected, and since e = s i=1 w i Proposition 1.3 yields the diameter bound.…”

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

“…This type of graph is often constructed in a more general setting applying to schemes (e.g. [BBV15]). This definition follows the definition of Hochster and Huneke [HH94] and is equivalent to other definitions, (e.g.…”

“…This definition follows the definition of Hochster and Huneke [HH94] and is equivalent to other definitions, (e.g. [BBV15]). In this paper we consider dual graphs of Stanley-Reisner rings.…”