THE SMOOTHNESS AND THE DIMENSION OF PGOR(H) AND OF OTHER STRATA OF THE PUNCTUAL HILBERT SCHEME.

Jan O. Kleppe

Mathematics Subject Classification (1991): 14C05, 13D10, 14B12, 13D03

Let PGor(H) be the scheme of all graded Gorenstein Artin R = k[X₁,X₂,X₃]-algebras A = R/I with Hilbert function H, endowed with its usual catalecticant structure. In this note we prove that PGor(H) is a smooth irreducible scheme and we compute its dimension. As a corollary we prove some conjectures of Geramita, Pucci and Shin on the Hilbert function of R/I². We also prove the smoothness and compute the dimension of ZGor(H) of all (not necessarily graded) Gorenstein quotients of k[[X₁,X₂,..,X_s]] with Hilbert function H at a graded Gorenstein quotient R -> A, leading to a criterion for A to be non-alignable.