Beitr\ EMIS ELibM Electronic Journals Beiträge zur Algebra und Geometrie
Contributions to Algebra and Geometry
Vol. 48, No. 1, pp. 141-150 (2007)

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A note on Rees algebras and the MFMC property

Isidoro Gitler, Carlos E. Valencia and Rafael H. Villarreal

Departamento de Matemáticas, Centro de Investigación y de Estudios Avanzados del IPN, Apartado Postal 14--740, 07000 México City, D.F., e-mail:

Abstract: We study irreducible representations of Rees cones and characterize the max-flow min-cut property of clutters in terms of the normality of Rees algebras and the integrality of certain polyhedra. Then we present some applications to combinatorial optimization and commutative algebra. As a byproduct we obtain an effective method, based on the program Normaliz\/ [B], to determine whether a given clutter satisfies the max-flow min-cut property. Let $\cal C$ be a clutter and let $I$ be its edge ideal. We prove that $\cal C$ has the max-flow min-cut property if and only if $I$ is normally torsion free, that is, $I^i=I^{(i)}$ for all $i\geq 1$, where $I^{(i)}$ is the $i$-th symbolic power of $I$. [B] Bruns, W.; Koch, R.: Normaliz -- a program for computing normalizations of affine semigroups. 1998. Available via anonymous ftp from ftp.mathematik.Uni-Osnabrueck.DE/pub/osm/kommalg/software

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Electronic version published on: 14 May 2007. This page was last modified: 27 Jan 2010.

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