Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

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



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
Published electronically: March 1, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • [1] Jeaman Ahn and Yong Su Shin, The minimal free resolution of a star-configuration in $ \mathbb{P}^n$ and the weak Lefschetz property, J. Korean Math. Soc. 49 (2012), no. 2, 405-417. MR 2933606,
  • [2] Jeaman Ahn and Yong Su Shin, The minimal free resolution of a fat star-configuration in $ \mathbb{P}^n$, Algebra Colloq. 21 (2014), no. 1, 157-166. MR 3163370,
  • [3] Thomas Bauer, Sandra Di Rocco, Brian Harbourne, Michał Kapustka, Andreas Knutsen, Wioletta Syzdek, and Tomasz Szemberg, A primer on Seshadri constants, Interactions of classical and numerical algebraic geometry, Contemp. Math., vol. 496, Amer. Math. Soc., Providence, RI, 2009, pp. 33-70. MR 2555949,
  • [4] Cristiano Bocci, Susan Cooper, Elena Guardo, Brian Harbourne, Mike Janssen, Uwe Nagel, Alexandra Seceleanu, Adam Van Tuyl, and Thanh Vu, The Waldschmidt constant for squarefree monomial ideals, J. Algebraic Combin. 44 (2016), no. 4, 875-904. MR 3566223,
  • [5] Cristiano Bocci and Brian Harbourne, Comparing powers and symbolic powers of ideals, J. Algebraic Geom. 19 (2010), no. 3, 399-417. MR 2629595,
  • [6] Enrico Carlini, Luca Chiantini, and Anthony V. Geramita, Complete intersections on general hypersurfaces, Special volume in honor of Melvin Hochster, Michigan Math. J. 57 (2008), 121-136. MR 2492444,
  • [7] Enrico Carlini, Elena Guardo, and Adam Van Tuyl, Star configurations on generic hypersurfaces, J. Algebra 407 (2014), 1-20. MR 3197149,
  • [8] Maria Virginia Catalisano, Anthony V. Geramita, Alessandro Gimigliano, and Yong-Su Shin, The secant line variety to the varieties of reducible plane curves, Ann. Mat. Pura Appl. (4) 195 (2016), no. 2, 423-443. MR 3476681,
  • [9] M. V. Catalisano, A. V. Geramita, A. Gimigliano, B. Harbourne, J. Migliore, U. Nagel, and Y. S. Shin, Secant varieties of the varieties of reducible hypersurfaces in $ \mathbb{P}^n$, preprint, 2015, arXiv:1502.00167.
  • [10] Susan Cooper, Brian Harbourne, and Zach Teitler, Combinatorial bounds on Hilbert functions of fat points in projective space, J. Pure Appl. Algebra 215 (2011), no. 9, 2165-2179. MR 2786607,
  • [11] Alberto Corso and Uwe Nagel, Specializations of Ferrers ideals, J. Algebraic Combin. 28 (2008), no. 3, 425-437. MR 2438922,
  • [12] Marcin Dumnicki, Brian Harbourne, Tomasz Szemberg, and Halszka Tutaj-Gasińska, Linear subspaces, symbolic powers and Nagata type conjectures, Adv. Math. 252 (2014), 471-491. MR 3144238,
  • [13] John A. Eagon and Victor Reiner, Resolutions of Stanley-Reisner rings and Alexander duality, J. Pure Appl. Algebra 130 (1998), no. 3, 265-275. MR 1633767,
  • [14] Christopher A. Francisco, Tetrahedral curves via graphs and Alexander duality, J. Pure Appl. Algebra 212 (2008), no. 2, 364-375. MR 2357338,
  • [15] Christopher A. Francisco, Juan C. Migliore, and Uwe Nagel, On the componentwise linearity and the minimal free resolution of a tetrahedral curve, J. Algebra 299 (2006), no. 2, 535-569. MR 2228326,
  • [16] A. V. Geramita, B. Harbourne, and J. Migliore, Star configurations in $ \mathbb{P}^n$, J. Algebra 376 (2013), 279-299. MR 3003727,
  • [17] A. V. Geramita, J. Migliore, and L. Sabourin, On the first infinitesimal neighborhood of a linear configuration of points in $ \mathbb{P}^2$, J. Algebra 298 (2006), no. 2, 563-611. MR 2217628,
  • [18] Elena Guardo, Brian Harbourne, and Adam Van Tuyl, Asymptotic resurgences for ideals of positive dimensional subschemes of projective space, Adv. Math. 246 (2013), 114-127. MR 3091802,
  • [19] Jan O. Kleppe, Juan C. Migliore, Rosa Miró-Roig, Uwe Nagel, and Chris Peterson, Gorenstein liaison, complete intersection liaison invariants and unobstructedness, Mem. Amer. Math. Soc. 154 (2001), no. 732, viii+116. MR 1848976,
  • [20] Magdalena Lampa-Baczyńska and Grzegorz Malara, On the containment hierarchy for simplicial ideals, J. Pure Appl. Algebra 219 (2015), no. 12, 5402-5412. MR 3390029,
  • [21] Carmelo Mammana, Sulla varietà delle curve algebriche piane spezzate in un dato modo, Ann. Scuola Norm. Super. Pisa (3) 8 (1954), 53-75 (Italian). MR 0064429
  • [22] Hideyuki Matsumura, Commutative ring theory, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1989. Translated from the Japanese by M. Reid. MR 1011461
  • [23] J. Migliore and U. Nagel, Tetrahedral curves, Int. Math. Res. Not. 15 (2005), 899-939. MR 2147092,
  • [24] Nguyen Cong Minh and Ngo Viet Trung, Cohen-Macaulayness of powers of two-dimensional squarefree monomial ideals, J. Algebra 322 (2009), no. 12, 4219-4227. MR 2558862,
  • [25] Nguyen Cong Minh and Ngo Viet Trung, Cohen-Macaulayness of monomial ideals and symbolic powers of Stanley-Reisner ideals, Adv. Math. 226 (2011), no. 2, 1285-1306. MR 2737785,
  • [26] Uwe Nagel and Victor Reiner, Betti numbers of monomial ideals and shifted skew shapes, Electron. J. Combin. 16 (2009), no. 2, Special volume in honor of Anders Bjorner, Research Paper 3, 59. MR 2515766
  • [27] Uwe Nagel and Tim Römer, Glicci simplicial complexes, J. Pure Appl. Algebra 212 (2008), no. 10, 2250-2258. MR 2426505,
  • [28] James G. Oxley, Matroid theory, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1992. MR 1207587
  • [29] Jung Pil Park and Yong-Su Shin, The minimal free graded resolution of a star-configuration in $ \mathbb{P}^n$, J. Pure Appl. Algebra 219 (2015), no. 6, 2124-2133. MR 3299722,
  • [30] Philip William Schwartau, Liaison addition and monomial ideals, ProQuest LLC, Ann Arbor, MI Thesis (Ph.D.)-Brandeis University, 1982. MR 2632011
  • [31] Naoki Terai and Ngo Viet Trung, Cohen-Macaulayness of large powers of Stanley-Reisner ideals, Adv. Math. 229 (2012), no. 2, 711-730. MR 2855076,
  • [32] Damiano Testa, Anthony Várilly-Alvarado, and Mauricio Velasco, Big rational surfaces, Math. Ann. 351 (2011), no. 1, 95-107. MR 2824848,
  • [33] Matteo Varbaro, Symbolic powers and matroids, Proc. Amer. Math. Soc. 139 (2011), no. 7, 2357-2366. MR 2784800,

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 14N20, 14M05, 05B35, 13F55, 05E40, 13D02, 13C40

Retrieve articles in all journals with MSC (2010): 14N20, 14M05, 05B35, 13F55, 05E40, 13D02, 13C40

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

B. Harbourne
Affiliation: Department of Mathematics, University of Nebraska, Lincoln, Nebraska 68588-0130

J. Migliore
Affiliation: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556

U. Nagel
Affiliation: Department of Mathematics, University of Kentucky, 715 Patterson Office Tower, Lexington, Kentucky 40506-0027

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

American Mathematical Society