Regulating Hartshorne's connectedness theorem

TL;DR

This paper provides a quantitative version of Hartshorne's connectedness theorem using Castelnuovo--Mumford regularity, establishing bounds on the dual graph's connectivity and constructing examples with minimal regularity.

Contribution

It introduces a regularity-based bound on the connectivity of the dual graph and constructs sharp examples, also linking graph properties to algebraic regularity measures.

Findings

The dual graph's connectivity is bounded by a function of regularities.

Every connected graph can be realized as the dual graph of a suitable projective curve.

The regularity of a curve is at most the sum of the regularities of its components.

Abstract

A classical theorem by Hartshorne states that the dual graph of any arithmetically Cohen--Macaulay projective scheme is connected. We give a quantitative version of Hartshorne's result, in terms of Castelnuovo--Mumford regularity. If $X \subset \mathbb{P}^n$ is an arithmetically Gorenstein projective scheme of regularity $r+1$, and if every irreducible component of $X$ has regularity $\le r'$, we show that the dual graph of $X$ is $\lfloor{\frac{r+r'-1}{r'}}\rfloor$-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 $\le 3$, 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.