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Matroid configurations and symbolic powers of their ideals


Authors: A. V. Geramita, B. Harbourne, J. Migliore and U. Nagel
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
Published electronically: March 1, 2017
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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
Email: Anthony.Geramita@gmail.com

B. Harbourne
Affiliation: Department of Mathematics, University of Nebraska, Lincoln, Nebraska 68588-0130
Email: bharbour@math.unl.edu

J. Migliore
Affiliation: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556
Email: migliore.1@nd.edu

U. Nagel
Affiliation: Department of Mathematics, University of Kentucky, 715 Patterson Office Tower, Lexington, Kentucky 40506-0027
Email: uwe.nagel@uky.edu

DOI: https://doi.org/10.1090/tran/6874
Keywords: Arithmetically Cohen-Macaulay, Hilbert function, hypergraph, hypersurface configuration, linkage, matroid, monomial ideal, resurgence, Stanley-Reisner ideal, star configuration, symbolic powers, tetrahedral curves, Waldschmidt constant.
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
Article copyright: © Copyright 2017 American Mathematical Society