PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 125, Number 4, April 1997, Pages 943–950 S 0002-9939(97)03956-7 REDUCED GORENSTEIN CODIMENSION THREE SUBSCHEMES OF PROJECTIVE SPACE ANTHONY V. GERAMITA AND JUAN C. MIGLIORE (Communicated by Eric M. Friedlander) Abstract. It is known, from work of Diesel, which graded Betti numbers are possible for Artinian Gorenstein height three ideals. In this paper we show that any such set of graded Betti numbers in fact occurs for a reduced set of points in P 3 ,a stick figure in P 4 , or more generally, a good linear configuration in P n . Consequently, any Gorenstein codimension three scheme specializes to such a “nice” configuration, preserving the graded Betti numbers in the process. This is the codimension three Gorenstein analog of a classical result of arithmetically Cohen-Macaulay codimension two schemes. 1. Introduction There are a number of well-known results on arithmetically Cohen-Macaulay (aCM) codimension two subschemes of projective space which fairly recently have been shown to have striking analogs for codimension three arithmetically Gorenstein subschemes. This paper gives a new analog to a classical result. The starting point for the modern results on codimension three Gorenstein schemes is the well-known structure theorem of D. Buchsbaum and D. Eisenbud [5]. There, the authors showed that there is one matrix which determines the res- olution, just as is done by the Hilbert-Burch matrix in codimension two. Then R. Stanley [24] showed that a symmetric sequence of integers is the Hilbert function of an Artinian codimension three Gorenstein algebra if and only if the first difference of the “first half” is the Hilbert function of a codimension two (aCM) Artinian algebra. These two results point the way toward a series of results for Gorenstein codimension three schemes, based on the defining matrix or on the Hilbert function, that are exactly analogous to standard results for the aCM codimension two case. We recall some of these. G. Ellingsrud [11] proved that the family of codimension two aCM schemes with the same Hilbert function is irreducible. It is also known that the family of codimension two aCM schemes with the same Hilbert function and degrees of generating sets for the homogeneous ideal is irreducible (cf. for instance [4], [7]). Both of these results have been proved for the Gorenstein codimension three case by S. Diesel [10]. Ellingsrud also proved that every aCM codimension two scheme Received by the editors July 24, 1995. 1991 Mathematics Subject Classification. Primary 14M05, 14C05, 13D02. c 1997 American Mathematical Society 943 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use