Combinatorial Bounds on Hilbert Functions of Fat Points in Projective Space

Susan Cooper, California Polytechnic State University - San Luis Obispo
Brian Harbourne, University of Nebraska-Lincoln
Zach Teitler, Boise State University

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NOTICE: This is the author’s version of a work accepted for publication by Elsevier. Changes resulting from the publishing process, including peer review, editing, corrections, structural formatting and other quality control mechanisms, may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. The definitive version has been published in Journal of Pure and Applied Algebra, February 2011. DOI: 10.1016/j.jpaa.2010.12.006

Abstract

We study Hilbert functions of certain non-reduced schemes A supported at finite sets of points in P^N, in particular, fat point schemes. We give combinatorially defined upper and lower bounds for the Hilbert function of A using nothing more than the multiplicities of the points and information about which subsets of the points are linearly dependent. When N = 2, we give these bounds explicitly and we give a sufficient criterion for the upper and lower bounds to be equal. When this criterion is satisfied, we give both a simple formula for the Hilbert function and combinatorially defined upper and lower bounds on the graded Betti numbers for the ideal I_A defining A, generalizing results of Geramita-Migliore-Sabourin [GMS]. We obtain the exact Hilbert functions and graded Betti numbers for many families of examples, interesting combinatorially, geometrically, and algebraically. Our method works in any characteristic.