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On 0-dimensional complete intersections. (English) Zbl 0741.14030
Given a 0-dimensional subscheme \(Y\) of \(\mathbb{P}^ d\), the paper discusses ways to characterize whether \(Y\) is a complete intersection, using numerical and geometrical properties of the embedding \(Y\subseteq\mathbb{P}^ d\). — In section 2 the well-known characterization of 0-dimensional complete intersections in \(\mathbb{P}^ 2\) by the symmetry of their Hilbert function and the Cayley-Bacharach property is shown to yield a characterization of 0-dimensional arithmetically Gorenstein schemes in general. This is accomplished by a careful study of the Cayley-Bacharach property using traces of the affine and projective coordinate rings of \(Y\).
For \(d\geq 3\) these invariants do not suffice anymore to characterize complete intersections. In section 3 the 0-dimensional zeroschemes of regular sections of certain vector bundles of rank 3 on \(\mathbb{P}^ 3\), called vector bundles with very good sections, are characterized. This and a suitable version of Horrocks’ splitting criterion are then used in section 4 to find the desired additional conditions for a characterization of 0-dimensional complete intersections in \(\mathbb{P}^ 3\). All characterizations are illustrated by examples.

MSC:
14M10 Complete intersections
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
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