Characterizations of zero-dimensional complete intersections
Determining when a 0-dimensional scheme in projective space \(\mathbb P^n_K\) (where \(K\) is a field) which is the complete intersection of \(n\) hypersurfaces has played an important role in both Algebraic Geometry and Commutative Algebra. For example, in projective 2-space, work of \textit{E. Davis} and \textit{P. Maroscia} [Lect. Notes Math. 1092, 253--269 (1984; Zbl 0556.14025)] provided a characterization of such schemes using the Hilbert function symmetry and the Cayley-Bacharach property. In [\textit{E. D. Davis} et al., Proc. Am. Math. Soc. 93, 593--597 (1985; Zbl 0575.14040)] and [\textit{M. Kreuzer}, Math. Ann. 292, No. 1, 43--58 (1992; Zbl 0741.14030)], it was then shown that this also provides a characterization for arithmetically Gorenstein schemes. There have also been characterizations of 0-dimensional local complete intersections using the Kähler different (see [\textit{G. Scheja} and \textit{U. Storch}, J. Reine Angew. Math. 278/279, 174--190 (1975; Zbl 0316.13003)]). In this paper the authors uses the Kähler and Dedekind differents of \(R/K[x_0]\) and \(R/K\) to further the characterizations of 0-dimensional complete intersections. A sample characterization is the following theorem: Theorem. Let \(\mathbb X\) be a smooth 0-dimensional subscheme of \(\mathbb P^n_K\). Then \(\mathbb X\) is a complete intersection if and only if \(\mathbb X\) is a Cayley-Bacharach scheme and the Hilbert function of the Kähler different of \(\mathbb X\) in degree \(r_{\mathbb X}\) is non-zero, where \(r_{\mathbb X}\) is the regularity index of the Hilbert function of \(\mathbb X\). Most of the characterizations can be found in Section 5 of the paper. Indeed, in this section the authors present three characterizations: one generalizing the work of Scheja and Storch; one using a value of the Hilbert function of the Kähler different of the scheme to separate the classes of complete intersections from arithmetically Gorenstein schemes; and one is the above stated theorem. Moreover, using the Dedekind different rather than the Kähler different, the authors characterize 0-dimensional arithmetically Gorenstein schemes. One particular gem of this paper is its useful exposition with many included necessary definitions and references to the literature.
scientific article; zbMATH DE number 6695680
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