Pascal's Triangle and Hilbert Functions of Points

Sindi Sabourin
(Queens University)

We define the Hilbert function of a finite set of points in projective
n-space.  We then state the result of Geramita, Maroscia and Roberts which
characterizes the Hilbert functions of points in terms of Macaulay's
O-sequences.  For every possible Hilbert function, they construct a set of
points having that Hilbert function.  Their constructions, called
k-configurations, have several properties of interest in themselves.  For
example, their Hilbert function, minimal free resolution and the degrees
of each point are easily determined.  We generalize k-configurations to
constructions which preserve these properties.  Furthermore, since
O-sequences are defined only in terms of the binomial coefficients, we are
able to use a generalized Pascal's triangle to generalize Macaulay's
O-sequences.  We use this generalization to characterize the Hilbert
functions of our genereralized k-configurations.