We study Hilbert functions of certain non-reduced schemes A supported at
finite sets of points in projective space, 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 defining A, generalizing
results of Geramita-Migliore-Sabourin (2006). 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. AWK scripts implementing our results can be obtained at
http://www.math.unl.edu/~bharbourne1/CHT/Example.html .Comment: 23 pages; changes have been made following suggestions of the
referee; explicit statements are now included for dimensions greater than 2,
hence the title no longer mentions the plane; however the content is largely
the same as in the previous version; this version is to appear in the Journal
of Pure and Applied Algebr
