### 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 (Formula presented.) is an arithmetically Gorenstein projective scheme of regularity (Formula presented.), and if every irreducible component of X has regularity (Formula presented.), we show that the dual graph of X is (Formula presented.)-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 (Formula presented.)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.

Original language | English (US) |
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Pages (from-to) | 1-18 |

Number of pages | 18 |

Journal | Journal of Algebraic Combinatorics |

DOIs | |

State | Accepted/In press - Feb 27 2017 |

### Keywords

- Arithmetically Gorenstein arrangements
- Balinski’s theorem
- Castelnuovo–Mumford regularity
- Dual graphs
- k-connectivity

### ASJC Scopus subject areas

- Algebra and Number Theory
- Discrete Mathematics and Combinatorics

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## Cite this

*Journal of Algebraic Combinatorics*, 1-18. https://doi.org/10.1007/s10801-017-0744-8