Regulating Hartshorne’s connectedness theorem

Bruno Benedetti, Barbara Bolognese, Matteo Varbaro

Research output: Contribution to journalArticlepeer-review

4 Scopus citations


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⊂ Pn 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 languageEnglish (US)
Pages (from-to)33-50
Number of pages18
JournalJournal of Algebraic Combinatorics
Issue number1
StatePublished - Aug 1 2017


  • 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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