The Waldschmidt constant for squarefree monomial ideals

Cristiano Bocci; Susan Cooper; Elena Guardo; Brian Harbourne; Mike Janssen; Uwe Nagel; Alexandra Seceleanu; Adam Van Tuyl; Thanh Vu

doi:10.1007/s10801-016-0693-7

The Waldschmidt constant for squarefree monomial ideals

Cristiano Bocci, Susan Cooper, Elena Guardo, Brian Harbourne, Mike Janssen, Uwe Nagel, Alexandra Seceleanu, Adam Van Tuyl, Thanh Vu

Mathematics

Research output: Contribution to journal › Article › peer-review

52 Scopus citations

Abstract

Given a squarefree monomial ideal I⊆ R= k[x₁, … , x_n] , we show that α^ (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 α^ (I) in terms of the fractional chromatic number of a hypergraph also constructed from the primary decomposition of I. Moreover, expressing α^ (I) as the solution to a linear program enables us to prove a Chudnovsky-like lower bound on α^ (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 Pⁿ with few components compared to n, and we compute the Waldschmidt constant for the Stanley–Reisner ideal of a uniform matroid.

Original language	English
Pages (from-to)	875-904
Number of pages	30
Journal	Journal of Algebraic Combinatorics
Volume	44
Issue number	4
DOIs	https://doi.org/10.1007/s10801-016-0693-7
State	Published - Dec 1 2016

Bibliographical note

Publisher Copyright:
© 2016, Springer Science+Business Media New York.

Funding

This project was started at the Mathematisches Forschungsinstitut Oberwolfach (MFO) as part of the mini-workshop \u201CIdeals of Linear Subspaces, Their Symbolic Powers and Waring Problems\u201D 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\u2019s NSF Grant DMS-1049268, \u201CNSF Junior Oberwolfach Fellows.\u201D Nagel was partially supported by the Simons Foundation under Grant No. 317096. Van Tuyl acknowledges the financial support provided by NSERC.

Funders	Funder number
PRIN
Natural Sciences and Engineering Research Council of Canada
Istituto Nazionale di Alta Matematica "Francesco Severi"
Gruppo Nazionale per le Strutture Algebriche, Geometriche e le loro Applicazioni
Dordt College
U.S. Department of Energy Chinese Academy of Sciences Guangzhou Municipal Science and Technology Project Oak Ridge National Laboratory Extreme Science and Engineering Discovery Environment National Science Foundation National Energy Research Scientific Computing Center National Natural Science Foundation of China	DMS-1049268, 1601024
U.S. Department of Energy Chinese Academy of Sciences Guangzhou Municipal Science and Technology Project Oak Ridge National Laboratory Extreme Science and Engineering Discovery Environment National Science Foundation National Energy Research Scientific Computing Center National Natural Science Foundation of China
National Security Agency	H98230-13-1-0213
National Security Agency
Simons Foundation	317096
Simons Foundation

Keywords

Fractional chromatic number
Graphs
Hypergraphs
Linear programming
Monomial ideals
Resurgence
Symbolic powers
Waldschmidt constant

ASJC Scopus subject areas

Algebra and Number Theory
Discrete Mathematics and Combinatorics

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10.1007/s10801-016-0693-7

Cite this

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title = "The Waldschmidt constant for squarefree monomial ideals",

abstract = "Given a squarefree monomial ideal I⊆ R= k[x1, … , xn] , we show that α\textasciicircum{} (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 α\textasciicircum{} (I) in terms of the fractional chromatic number of a hypergraph also constructed from the primary decomposition of I. Moreover, expressing α\textasciicircum{} (I) as the solution to a linear program enables us to prove a Chudnovsky-like lower bound on α\textasciicircum{} (I) , thus verifying a conjecture of Cooper–Embree–H{\`a}–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 Pn 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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author = "Cristiano Bocci and Susan Cooper and Elena Guardo and Brian Harbourne and Mike Janssen and Uwe Nagel and Alexandra Seceleanu and Tuyl, \{Adam Van\} and Thanh Vu",

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AU - Cooper, Susan

AU - Guardo, Elena

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AU - Nagel, Uwe

AU - Seceleanu, Alexandra

AU - Tuyl, Adam Van

AU - Vu, Thanh

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N2 - Given a squarefree monomial ideal I⊆ R= k[x1, … , xn] , we show that α^ (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 α^ (I) in terms of the fractional chromatic number of a hypergraph also constructed from the primary decomposition of I. Moreover, expressing α^ (I) as the solution to a linear program enables us to prove a Chudnovsky-like lower bound on α^ (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 Pn with few components compared to n, and we compute the Waldschmidt constant for the Stanley–Reisner ideal of a uniform matroid.

AB - Given a squarefree monomial ideal I⊆ R= k[x1, … , xn] , we show that α^ (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 α^ (I) in terms of the fractional chromatic number of a hypergraph also constructed from the primary decomposition of I. Moreover, expressing α^ (I) as the solution to a linear program enables us to prove a Chudnovsky-like lower bound on α^ (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 Pn 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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The Waldschmidt constant for squarefree monomial ideals

Abstract

Bibliographical note

Funding

Keywords

ASJC Scopus subject areas

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