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Transactions of the American Mathematical Society

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ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Matroid configurations and symbolic powers of their ideals
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by A. V. Geramita, B. Harbourne, J. Migliore and U. Nagel PDF
Trans. Amer. Math. Soc. 369 (2017), 7049-7066 Request permission

Abstract:

Star configurations are certain unions of linear subspaces of projective space that have been studied extensively. We develop a framework for studying a substantial generalization, which we call matroid configurations, whose ideals generalize Stanley-Reisner ideals of matroids. Such a matroid configuration is a union of complete intersections of a fixed codimension. Relating these to the Stanley-Reisner ideals of matroids and using methods of liaison theory allows us, in particular, to describe the Hilbert function and minimal generators of the ideal of, what we call, a hypersurface configuration. We also establish that the symbolic powers of the ideal of any matroid configuration are Cohen-Macaulay. As applications, we study ideals coming from certain complete hypergraphs and ideals derived from tetrahedral curves. We also consider Waldschmidt constants and resurgences. In particular, we determine the resurgence of any star configuration and many hypersurface configurations. Previously, the only non-trivial cases for which the resurgence was known were certain monomial ideals and ideals of finite sets of points. Finally, we point out a connection to secant varieties of varieties of reducible forms.
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Additional Information
  • A. V. Geramita
  • Affiliation: Department of Mathematics and Statistics, Queen’s University, Kingston, Ontario, Canada – and – Dipartimento di Matematica, Università di Genova, Genoa, Italy
  • MR Author ID: 72575
  • Email: Anthony.Geramita@gmail.com
  • B. Harbourne
  • Affiliation: Department of Mathematics, University of Nebraska, Lincoln, Nebraska 68588-0130
  • MR Author ID: 217048
  • Email: bharbour@math.unl.edu
  • J. Migliore
  • Affiliation: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556
  • MR Author ID: 124490
  • ORCID: 0000-0001-5528-4520
  • Email: migliore.1@nd.edu
  • U. Nagel
  • Affiliation: Department of Mathematics, University of Kentucky, 715 Patterson Office Tower, Lexington, Kentucky 40506-0027
  • MR Author ID: 248652
  • Email: uwe.nagel@uky.edu
  • Received by editor(s): July 1, 2015
  • Received by editor(s) in revised form: October 16, 2015
  • Published electronically: March 1, 2017
  • Additional Notes: While this paper was being processed for publication, Tony Geramita passed away. On behalf of Tony’s many friends and colleagues from all walks of life, the three remaining authors dedicate this paper to him.

  • Dedicated: In fond memory of A. V. Geramita, 1942–2016
  • © Copyright 2017 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 369 (2017), 7049-7066
  • MSC (2010): Primary 14N20, 14M05, 05B35; Secondary 13F55, 05E40, 13D02, 13C40
  • DOI: https://doi.org/10.1090/tran/6874
  • MathSciNet review: 3683102