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

http://dx.doi.org/10.4153/CJM-2001-037-3
Canad. J. Math. 53(2001), 923-943

  Published:2001-10-01
 Printed: Oct 2001

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.