Generalized k-Configurations

Abstract

AbstractIn this paper, we find configurations of points in n-dimensional projective space (Pn) which simultaneously generalize both k-configurations and reduced 0-dimensional complete intersections. Recall that k-configurations in P2 are disjoint unions of distinct points on lines and in Pn are inductively disjoint unions of k-configurations on hyperplanes, subject to certain conditions. Furthermore, the Hilbert function of a k-configuration is determined from those of the smaller k-configurations. We call our generalized constructions kD-configurations, where D = {d1, … , dr} (a set of r positive integers with repetition allowed) is the type of a given complete intersection in Pn. We show that the Hilbert function of any kD-configuration can be obtained from those of smaller kD-configurations. We then provide applications of this result in two different directions, both of which are motivated by corresponding results about k-configurations.