### 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) |
---|---|

Pages (from-to) | 1-18 |

Number of pages | 18 |

Journal | Journal of Algebraic Combinatorics |

DOIs | |

State | Accepted/In press - Feb 27 2017 |

### Fingerprint

### 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

### Cite this

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

**Regulating Hartshorne’s connectedness theorem.** / Benedetti, Bruno; Bolognese, Barbara; Varbaro, Matteo.

Research output: Contribution to journal › Article

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

}

TY - JOUR

T1 - Regulating Hartshorne’s connectedness theorem

AU - Benedetti, Bruno

AU - Bolognese, Barbara

AU - Varbaro, Matteo

PY - 2017/2/27

Y1 - 2017/2/27

N2 - 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.

AB - 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.

KW - Arithmetically Gorenstein arrangements

KW - Balinski’s theorem

KW - Castelnuovo–Mumford regularity

KW - Dual graphs

KW - k-connectivity

UR - http://www.scopus.com/inward/record.url?scp=85014048060&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85014048060&partnerID=8YFLogxK

U2 - 10.1007/s10801-017-0744-8

DO - 10.1007/s10801-017-0744-8

M3 - Article

AN - SCOPUS:85014048060

SP - 1

EP - 18

JO - Journal of Algebraic Combinatorics

JF - Journal of Algebraic Combinatorics

SN - 0925-9899

ER -