Reduced arithmetically Gorenstein schemes and simplical polytopes with maximal Betti numbers

Juan C. Migliore, Uwe Nagel

Research output: Contribution to journalArticlepeer-review

45 Scopus citations

Abstract

An SI-sequence is a finite sequence of positive integers which is symmetric, unimodal and satisfies a certain growth condition. These are known to correspond precisely to the possible Hilbert functions of graded Artinian Gorenstein algebras with the weak Lefschetz property, a property shared by a nonempty open set of the family of all graded Artinian Gorenstein algebras having a fixed Hilbert function that is an SI sequence. Starting with an arbitrary SI-sequence, we construct a reduced, arithmetically Gorenstein configuration G of linear varieties of arbitrary dimension whose Artinian reduction has the given SI-sequence as Hilbert function and has the weak Lefschetz property. Furthermore, we show that G has maximal graded Betti numbers among all arithmetically Gorenstein subschemes of projective space whose Artinian reduction has the weak Lefschetz property and the given Hilbert function. As an application we show that over a field of characteristic zero every set of simplicial polytopes with fixed h-vector contains a polytope with maximal graded Betti numbers.

Original languageEnglish
Pages (from-to)1-63
Number of pages63
JournalAdvances in Mathematics
Volume180
Issue number1
DOIs
StatePublished - Dec 1 2003

Keywords

  • Arithmetically
  • Compressed algebra
  • Gorenstein ideal
  • Gorenstein liaison
  • Gorenstein subscheme
  • Graded Betti numbers
  • Hilbert function
  • SI-sequence
  • Simplical polytopes
  • Weak Lefschetz property

ASJC Scopus subject areas

  • Mathematics (all)

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