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. e-mail:loc@maths.ed.ac.uk

University of Paderborn, Department of Mathematics and Computer Science, 33098 Paderborn, Germany, e-mail: thilop@uni-paderborn.de

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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