| Abstract:
A famous theorem due to F. S. Macaulay characterizes Hilbert
functions of quotients of polynomial rings. Adding further restrictions,
work of Geramita-Maroscia-Roberts gives us a characterization of the
Hilbert functions of reduced 0-dimensional schemes. These results have
played central roles in solving many interesting problems from Commutative
Algebra and Algebraic Geometry and hence it is natural to want a similar
characterization for non-reduced schemes called fat point schemes.
However, we are far from a complete characterization in the setting of fat
point schemes. In this talk we will look at a reduction procedure which
yields bounds (and sometimes a precise calculation!) of Hilbert functions
of fat points.
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