Abstract view
Decompositions of the Hilbert Function of a Set of Points in $\P^n$


Published:20011001
Printed: Oct 2001
Anthony V. Geramita
Tadahito Harima
Yong Su Shin
Abstract
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 $\alpha
= d_s + d_{s1} + \cdots + d_1$ (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.