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

34 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

Funding Information:
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.

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

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^ (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–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.",

keywords = "Fractional chromatic number, Graphs, Hypergraphs, Linear programming, Monomial ideals, Resurgence, Symbolic powers, Waldschmidt constant",

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",

note = "Funding Information: 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{\textquoteright}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. Publisher Copyright: {\textcopyright} 2016, Springer Science+Business Media New York.",

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AU - Cooper, Susan

AU - Guardo, Elena

AU - Harbourne, Brian

AU - Janssen, Mike

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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KW - Hypergraphs

KW - Linear programming

KW - Monomial ideals

KW - Resurgence

KW - Symbolic powers

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The Waldschmidt constant for squarefree monomial ideals

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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