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

Published online by Cambridge University Press:  20 November 2018

Anthony V. Geramita ,
Tadahito Harima  and
Yong Su Shin
Anthony V. Geramita
Affiliation:
Department of Mathematics, Queen's University, Kingston, Ontario, K7L 3N6 and Dipartimento di Matematica, Universitá di Genova, Genova, Italia. email: tony@mast.queensu.ca, geramita@dima.unige.it
Tadahito Harima
Affiliation:
Department of Management and Information Science, Shikoku University, Tokushima 771-11, Japan. email: harima@keiei.shikoku-u.ac.jp
Yong Su Shin
Affiliation:
Department of Mathematics, Sung Shin Women's University, 249-1, Dong Sun Dong 3Ka, Sung Buk Ku, Seoul, Korea 136-742. email: ysshin@cc.sungshin.ac.kr, ysshin@mast.queensu.ca
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Abstract

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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.

Keywords

13D4014M10
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 53 , Issue 5 , 01 October 2001 , pp. 923 - 943
Copyright
Copyright © Canadian Mathematical Society 2001

References

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