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.
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. http://dx.doi.org/10.1016/j.jpaa.2010.12.006