On the Hilbert function of graded algebras of dimension 0 (Q2782019)
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scientific article; zbMATH DE number 1727414
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | On the Hilbert function of graded algebras of dimension 0 |
scientific article; zbMATH DE number 1727414 |
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11 April 2002
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graded algebras
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0-dimensional Gorenstein ideals
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Hilbert function
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complete intersection
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Noether-Lefschetz locus
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0.92470115
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0.9149026
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On the Hilbert function of graded algebras of dimension 0 (English)
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We study graded algebras associated to 0-dimensional Gorenstein ideals \(I\subset\mathbb{C}[X_0,\dots,X_r]\) with fixed socle degree and such that \(I\) contains a complete intersection of fixed multidegree. We conjecture that among such algebras those minimizing the Hilbert function are some complete intersections. We prove an asymptotic version of this conjecture.NEWLINENEWLINENEWLINEOur result is an important case of a conjecture of \textit{D. Eisenbud}, \textit{M. Green} and \textit{J. Harris} [in: Journées Géom. Algébr., Orsay 1992, Astérisque 218, 187-202 (1993; Zbl 0819.14001)]. In Hodge theory, it enables us to compute the biggest component of the Noether-Lefschetz locus of projective hypersurfaces of very high degree, in any dimension.
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