Document Type
Article
Publication Date
9-2011
Abstract
We study Hilbert functions of certain non-reduced schemes A supported at finite sets of points in PN, 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 IA 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.
Copyright Statement
This is an author-produced, peer-reviewed version of this article. © 2009, Elsevier. Licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License (https://creativecommons.org/licenses/by-nc-nd/4.0/). The final, definitive version of this document can be found online at Journal of Pure and Applied Algebra, doi: 10.1016/j.jpaa.2010.12.006
Publication Information
Cooper, Susan; Harbourne, Brian; and Teitler, Zach. (2011). "Combinatorial Bounds on Hilbert Functions of Fat Points in Projective Space". Journal of Pure and Applied Algebra, 215(9), 2165-2179. https://doi.org/10.1016/j.jpaa.2010.12.006