Generalized $k$-Configurations

[摘要] In this paper, we find configurations of points in $n$-dimensionalprojective space ($proj ^n$) which simultaneously generalize both$k$-configurations and reduced 0-dimensional complete intersections.Recall that $k$-configurations in $proj ^2$ are disjoint unions ofdistinct points on lines and in $proj ^n$ are inductively disjointunions of $k$-configurations on hyperplanes, subject to certainconditions. Furthermore, the Hilbert function of a $k$-configurationis determined from those of the smaller $k$-configurations. We callour generalized constructions $k_D$-configurations, where $D={ d_1,ldots ,d_r}$ (a set of $r$ positive integers with repetitionallowed) is the type of a given complete intersection in $proj ^n$.We show that the Hilbert function of any $k_D$-configuration can beobtained from those of smaller $k_D$-configurations. We then provideapplications of this result in two different directions, both of whichare motivated by corresponding results about $k$-configurations.

[学科分类] 数学（综合）