Beiträge zur Algebra und Geometrie / Contributions to Algebra and GeometryVol. 40, No. 1, pp. 229-242 (1999)

Deformation to the Normal Cone, Rationality and the Stückrad-Vogel Cycle

L. O'Carroll, T. Pruschke

Department of Mathematics and Statistics, University of Edinburgh, Edinburgh EH9 3JZ, Scotland, U.K.

University of Paderborn, Department of Mathematics and Computer Science, 33098 Paderborn, Germany, e-mail:

Abstract: The Stückrad-Vogel cycle of refined intersection theory collects together the irreducible components of the intersection cycle, together with their multiplicities, needed for a version of Bezout's Theorem applicable to the case of possibly improper intersection. The invariance of the Stückrad-Vogel cycle under deformation to the normal cone forms the cornerstone of van Gastel's (geometrical) proof of the coincidence of the refined intersection theories of Fulton-MacPherson and Stückrad-Vogel. Later on, Achilles and Manaresi gave an algebraic proof of a local version of this invariance first at a rational and subsequently at a general component of the cycle. Their approach used a detailed analysis of the behaviour of filter-regular sequences in associated graded rings. In related work, van Gastel showed the coincidence of the distinguished varieties of Fulton with the rational components of the Stückrad-Vogel cycle. Subsequently, Achilles and Manaresi obtained a local version of the fact that a rational component was necessarily distinguished in a slightly weak sense (see the introduction to Section 3), using an algebraic approach involving analytic spread.

The first aim of this paper is to present a natural and alternative independent approach to the work of Achilles and Manaresi by giving an algebraic proof of the general fact of the invariance of the Stückrad-Vogel cycle under deformation to the normal cone, using extended Rees rings as a natural deformation space (see Section 2), and by refining and generalizing somewhat their basic theorem which discusses the connections between distinguished and rational components. Indeed a possibly non-reduced version of this generalization is given, using previous work of O'Carroll and Qureshi on tensor products of fields (see Section 3). Section 4 is devoted to an example which illustrates a paradoxical aspect of our work.

The other aim of this paper is to revisit a classical result of Northcott and Rees which essentially states that the correct number of sufficiently general elements in an ideal of a local ring generates a minimal reduction; this is proved using the rather technical machinery of `resultant systems' (plural). The same approach is employed by Achilles and Manaresi in their work on rational components of the Stückrad-Vogel cycle. In the final section we sketch a clean geometrical proof of this classical result, using the old notion of `Cayley form' (singular) (also called `ground form' or `associated form'), which we hope is of interest.

Classification (MSC91): 14C17, 13H15, 13A30

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