Decompositions of the Hilbert Function of a Set of Points in Pn

Geramita, Anthony V. ; Harima, Tadahito ; Shin, Yong Su

Canadian journal of mathematics, Tome 53 (2001) no. 5, pp. 923-943

Résumé

Let $\mathbf{H}$ be the Hilbert function of some set of distinct points in ${{\mathbb{P}}^{n}}$ and let $\alpha \,=\,\alpha (\mathbf{H})$ be the least degree of a hypersurface of ${{\mathbb{P}}^{n}}$ containing these points. Write $\alpha ={{d}_{s}}+{{d}_{s-1}}+\cdot \cdot \cdot +{{d}_{1}}$ (where ${{d}_{i}}>0$ ). We canonically decompose $\mathbf{H}$ into $s$ other Hilbert functions $\text{H}\leftrightarrow \text{(}{{\text{H}'}_{s}}\text{,}...\text{,}{{\text{H}'}_{1}}\text{)}$ and show how to find sets of distinct points ${{\mathbb{Y}}_{s}},...,{{\mathbb{Y}}_{1}}$ , lying on reduced hypersurfaces of degrees ${{d}_{s}},...,{{d}_{1}}$ (respectively) such that the Hilbert function of ${{\mathbb{Y}}_{i}}$ is ${{\text{H'}}_{i}}$ and the Hilbert function of $\mathbb{Y}=\bigcup _{i=1}^{s}\,{{\mathbb{Y}}_{i}}$ is $\mathbf{H}$ . Some extremal properties of this canonical decomposition are also explored.

Détail

DOI : 10.4153/CJM-2001-037-3

Mots-clés : 13D40, 14M10

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