Blow-up algebras, determinantal ideals, and Dedekind-Mertens-like formulas

Alberto Corso, Uwe Nagel, Sonja Petrović, Cornelia Yuen

Research output: Contribution to journalArticlepeer-review

8 Scopus citations

Abstract

We investigate Rees algebras and special fiber rings obtained by blowing up specialized Ferrers ideals. This class of monomial ideals includes strongly stable monomial ideals generated in degree two and edge ideals of prominent classes of graphs. We identify the equations of these blow-up algebras. They generate determinantal ideals associated to subregions of a generic symmetric matrix, which may have holes. Exhibiting Gröbner bases for these ideals and using methods from Gorenstein liaison theory, we show that these determinantal rings are normal Cohen-Macaulay domains that are Koszul, that the initial ideals correspond to vertex decomposable simplicial complexes, and we determine their Hilbert functions and Castelnuovo-Mumford regularities. As a consequence, we find explicit minimal reductions for all Ferrers and many specialized Ferrers ideals, as well as their reduction numbers. These results can be viewed as extensions of the classical Dedekind-Mertens formula for the content of the product of two polynomials.

Original languageEnglish
Pages (from-to)799-830
Number of pages32
JournalForum Mathematicum
Volume29
Issue number4
DOIs
StatePublished - Jul 1 2017

Bibliographical note

Funding Information:
National Security Agency under grant numbers H98230-09-1-0032 and H98230-12-1-0247

Publisher Copyright:
© 2017 Walter de Gruyter GmbH, Berlin/Boston 2017.

Keywords

  • Blow-up algebras
  • Castelnuovo Mumford regularity
  • Cohen Macaulay algebra
  • Ferrers and threshold graphs
  • Gröbner basis
  • Hilbert function
  • determinantal ideal
  • liaison
  • quadratic and Koszul algebra
  • reductions
  • skew shapes
  • vertex-decomposability

ASJC Scopus subject areas

  • Mathematics (all)
  • Applied Mathematics

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