JOURNAL OF ALGEBRA 90, 528-555 (1984) The Ideal of Forms Vanishing at a Finite Set of Points in Ipn A. V. GERAMITA* Department of Mathematics and Statistics, Queen’s University, Kingston, Ontario, Canada AND P. MAROSCIA~ Ist. Mat. “G. Castelnuovo,” Universitd di Roma, Cittci Universitaria, 00100 Rotna, Italy Communicated by I. N. Herstein Received March 3, 1983 0. INTRODUCTION Let P , ,..., P, be points of iPn(k), with s 2 n 2 2 and k = k an algebraically closed field. Let Z = Id 0 Id+ I @ ... , with Id # (0), be the ideal of P, ,..., P, in k[x,,..., xn] and let v(Z) denote the minimal number of generators of I. In this paper we shall be mainly concerned with the following problems: PROBLEM (A). Does there exist, for any given II and s, a non-empty Zariski open set U,,, c (IP”)’ such that if P = (p, ,..., P,) E U,,, and Z is the ideal of P r ,..., P,, then v(Z) is a constant independent of P and explicitly computable in terms of n and s? PROBLEM (B). If such an open set U,,, exists, can one find some dense subsets of it which are “easily” identifiable in some concrete geometric or algebraic way? The motivation for this study comes from several sources: first, the very stimulating treatments of Macaulay’s examples given by Abhyankar [A] and *Supported in part by a grant from the Natural Science and Engineering Research Council of Canada under Grant A8488. + Supported by Consiglio Nazionale delle Ricerche. This author would like to thank the Mathematics and Statistics Department of Queen’s University, Kingston, Ontario, for their kind hospitality during the preparation of this work. 528 0021-8693/84 $3.00 Copyright 0 1984 by Academic Press, Inc. All rights of reproduction in any form reserved.