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Gorenstein algebras and the Cayley-Bacharach theorem. (English) Zbl 0575.14040
It is known, if not well-known, that the classical Cayley-Bacharach theorem for complete intersections in \({\mathbb{P}}^ 2\) is valid for 0- dimensional arithmetically Gorenstein subschemes of \({\mathbb{P}}^ n\). It is shown, more generally, that the result can be extended to 0-dimensional arithmetically Cohen-Macaulay subschemes of \({\mathbb{P}}^ n\) whose minimal Cohen-Macauley type is compatible with their Hilbert functions. The Gorenstein version of the Cayley-Bacharach theorem is deduced from a technical result which relates the Hilbert functions of linked subschemes. The note ends by showing that the 0-dimensional, arithmetically Gorenstein, reduced subschemes of \({\mathbb{P}}^ n\) are characterized by the validity of the Cayley-Bacharach theorem and the symmetry of the Hilbert function.
Reviewer: D.Kirby

14M05 Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal)
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
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