Regulating Hartshorne’s connectedness theorem

Bruno Benedetti, Barbara Bolognese, Matteo Varbaro

Research output: Contribution to journalArticle

3 Citations (Scopus)

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 languageEnglish (US)
Pages (from-to)1-18
Number of pages18
JournalJournal of Algebraic Combinatorics
DOIs
StateAccepted/In press - Feb 27 2017

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Connectedness
Regularity
Theorem
Dual Graph
Curve
Irreducible Components
Gorenstein
Local Ring
Graph in graph theory
Converse
Connected graph

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

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

In: Journal of Algebraic Combinatorics, 27.02.2017, p. 1-18.

Research output: Contribution to journalArticle

Benedetti, Bruno ; Bolognese, Barbara ; Varbaro, Matteo. / Regulating Hartshorne’s connectedness theorem. In: Journal of Algebraic Combinatorics. 2017 ; pp. 1-18.
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