Regulating Hartshorne’s connectedness theorem

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: 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 dual to 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 are: (1) every 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.