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The Waldschmidt constant for squarefree monomial ideals

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Abstract

Given a squarefree monomial ideal \(I \subseteq R =k[x_1,\ldots ,x_n]\), we show that \(\widehat{\alpha }(I)\), the Waldschmidt constant of I, can be expressed as the optimal solution to a linear program constructed from the primary decomposition of I. By applying results from fractional graph theory, we can then express \(\widehat{\alpha }(I)\) in terms of the fractional chromatic number of a hypergraph also constructed from the primary decomposition of I. Moreover, expressing \(\widehat{\alpha }(I)\) as the solution to a linear program enables us to prove a Chudnovsky-like lower bound on \(\widehat{\alpha }(I)\), thus verifying a conjecture of Cooper–Embree–Hà–Hoefel for monomial ideals in the squarefree case. As an application, we compute the Waldschmidt constant and the resurgence for some families of squarefree monomial ideals. For example, we determine both constants for unions of general linear subspaces of \(\mathbb P^n\) with few components compared to n, and we compute the Waldschmidt constant for the Stanley–Reisner ideal of a uniform matroid.

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Acknowledgments

This project was started at the Mathematisches Forschungsinstitut Oberwolfach (MFO) as part of the mini-workshop “Ideals of Linear Subspaces, Their Symbolic Powers and Waring Problems” organized by C. Bocci, E. Carlini, E. Guardo and B. Harbourne. All the authors thank the MFO for providing a stimulating environment. Bocci acknowledges the financial support provided by GNSAGA of INdAM. Guardo acknowledges the financial support provided by PRIN 2011. Harbourne was partially supported by NSA Grant Number H98230-13-1-0213. Janssen was partially supported by Dordt College. Janssen and Seceleanu received support from MFO’s NSF Grant DMS-1049268, “NSF Junior Oberwolfach Fellows.” Nagel was partially supported by the Simons Foundation under Grant No. 317096. Van Tuyl acknowledges the financial support provided by NSERC.

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Authors and Affiliations

  1. Department of Information Engineering and Mathematics, University of Siena, Via Roma, 56, Siena, Italy

    Cristiano Bocci

  2. Department of Mathematics, North Dakota State University, NDSU Dept #2750, PO Box 6050, Fargo, ND, 58108-6050, USA

    Susan Cooper

  3. Dipartimento di Matematica e Informatica, Università di Catania, Viale A. Doria, 6, 95100, Catania, Italy

    Elena Guardo

  4. Department of Mathematics, University of Nebraska, Lincoln, NE, 68588-0130, USA

    Brian Harbourne, Alexandra Seceleanu & Thanh Vu

  5. Department of Mathematics, Statistics, and Computer Science, Dordt College, Sioux Center, IA, 51250, USA

    Mike Janssen

  6. Department of Mathematics, University of Kentucky, 715 Patterson Office Tower, Lexington, KY, 40506-0027, USA

    Uwe Nagel

  7. Department of Mathematics and Statistics, McMaster University, Hamilton, ON, L8S 4L8, Canada

    Adam Van Tuyl

Authors
  1. Cristiano Bocci
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  2. Susan Cooper
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  3. Elena Guardo
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  4. Brian Harbourne
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  5. Mike Janssen
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  6. Uwe Nagel
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  7. Alexandra Seceleanu
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  8. Adam Van Tuyl
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  9. Thanh Vu
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Correspondence to Adam Van Tuyl.

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Bocci, C., Cooper, S., Guardo, E. et al. The Waldschmidt constant for squarefree monomial ideals. J Algebr Comb 44, 875–904 (2016). https://doi.org/10.1007/s10801-016-0693-7

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  • DOI: https://doi.org/10.1007/s10801-016-0693-7

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