Midwest Algebra, Geometry and their Interactions Conference MAGIC05 University of Notre Dame, Notre Dame October 7-11, 2005 Parameter spaces of quotients of k[x0, ..., xn] with fixed Hilbert function by Jan O. Kleppe, Oslo University College, Norway Abstract: The Hilbert function stratum, GradAlg(H), of the Hilbert scheme Hilbp(x)(Pn) with fixed Hilbert function and the corresponding parameter space in the Artinian case are studied with respect to dimension, smoothness and reducibility. In particular if B ® A are graded quotient of a polynomial ring and A is constructed from B in a well organized manner such as B is Cohen-Macaulay (CM) and A is the Gorenstein algebra obtained by taking regular sections of a twist of the dual of the canonical module (or of certain other maximal CM-modules), or A is given by "some truncation of B" then the construction takes an open neighborhood of (B) in GradAlg(HB) onto an open of (A) in GradAlg(HA) under some conditions. This gives us a tool to classify certain Gorenstein and level algebras of codimension (height) 3 (resp. 4) by using a corresponding classification of CM-algebras of codimension 2 (resp. 3). Moreover we report on GradAlg(HA) when A is the homogeneous coordinate ring of any Buchsbaum curve of diameter 1.