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1. CJM 2005 (vol 57 pp. 400)
Generalized $k$-Configurations In this paper, we find configurations of points in $n$-dimensional
projective 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 of
distinct points on lines and in $\proj ^n$ 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 $k_D$-configurations, where $D=\{ d_1,
\ldots ,d_r\}$ (a set of $r$ positive integers with repetition
allowed) is the type of a given complete intersection in $\proj ^n$.
We show that the Hilbert function of any $k_D$-configuration can be
obtained from those of smaller $k_D$-configurations. We then provide
applications of this result in two different directions, both of which
are motivated by corresponding results about $k$-configurations.
Categories:13D40, 14M10 |
2. CJM 2001 (vol 53 pp. 923)
Decompositions of the Hilbert Function of a Set of Points in $\P^n$ Let $\H$ be the Hilbert function of some set of distinct points
in $\P^n$ and let $\alpha = \alpha (\H)$ be the least degree
of a hypersurface of $\P^n$ containing these points. Write $\alpha
= d_s + d_{s-1} + \cdots + d_1$ (where $d_i > 0$). We canonically
decompose $\H$ into $s$ other Hilbert functions $\H
\leftrightarrow (\H_s^\prime, \dots, \H_1^\prime)$ and show
how to find sets of distinct points $\Y_s, \dots, \Y_1$,
lying on reduced hypersurfaces of degrees $d_s, \dots, d_1$
(respectively) such that the Hilbert function of $\Y_i$ is
$\H_i^\prime$ and the Hilbert function of $\Y = \bigcup_{i=1}^s
\Y_i$ is $\H$. Some extremal properties of this canonical
decomposition are also explored.
Categories:13D40, 14M10 |