Decompositions of the Hilbert function of a set of points in \(\mathbb P^n\) (Q5890371)

As the title suggests, the topic of this paper is the decomposition of the Hilbert function of a set of points in \(\mathbb P^n\) into Hilbert functions of simpler sets of points. A first result in this direction was shown by \textit{A. V. Geramita}, \textit{P. Maroscia} and \textit{L. Roberts} [J. Lond. Math. Soc. (2) 28, 443--452 (1983; Zbl 0535.13012)], where a decomposition was found the components of which are Hilbert functions of sets of points in codimension one linear subspaces of \(\mathbb P^n\).NEWLINENEWLINENEWLINEThe main result of the paper under review is a generalization of that decomposition. Let \(\mathbb X \subset \mathbb P^n\) be a set of points whose vanishing ideal has initial degree \(\alpha\), and let \(\alpha=d_1 +\cdots+ d_s\). Then the main theorem gives a canonical decomposition of the Hilbert function of \(\mathbb X\) into Hilbert functions of sets of points on hypersurfaces of \(\mathbb P^n\) of degree \(d_i\). As in the case of the linear decomposition, a simple formula describes the value of the Hilbert function of \(\mathbb X\) in terms of the values of the component Hilbert functions.NEWLINENEWLINENEWLINEThe component Hilbert functions have certain connections with each other. The authors explain those connections and show that any list of component Hilbert functions satisfying them can be recombined into a union Hilbert function. Furthermore, \(k\)-configurations of points are shown to have extremal properties with respect to this decomposition.NEWLINENEWLINENEWLINEThe paper is written in a clear and rather self-contained style. The main proof technique is the use of \(n\)-type vectors for coding the information contained in the Hilbert function of a set of points.