## 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⊂ 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′-1r′⌋-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 ≤ 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) |
---|---|

Pages (from-to) | 33-50 |

Number of pages | 18 |

Journal | Journal of Algebraic Combinatorics |

Volume | 46 |

Issue number | 1 |

DOIs | |

State | Published - Aug 1 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