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Analytic improper intersections. [Pt] 2, Deformation to an algebraic bicone and applications

Nowak K.

Bulletin of the Polish Academy of Sciences. Mathematics

2000

Vol. 48, no 2

131-140

This article, being the second part of our paper [7], continues to study the analytic intersection algorithm. We present a certain method of deformation of an analytic set to an algebraic bicone, and next we express the result of the analytic intersection algorithm and multiplicity for improper intersections as a degree sequence and the Samuel multiplicity of that bicone, respectively (cf. [6]). Also produced are many important consequences (cf. [6]), among others the coincidence between the intersection indices for analytic improper intersections defined by Tworzewski [12] and those defined by Achilles and Manaresi [1] (Corollary 3), the linear testing theorem (Corollary 6) or a generalization of the classical reduction theorem to the case of analytic improper intersections (Corollary 7) which ensures the canonical character of the diagonal procedure.

Analytic improper intersections. [Pt] 1, Deformation to the normal cone

Nowak K.

Bulletin of the Polish Academy of Sciences. Mathematics

2000

Vol. 48, no 2

121-130

In this paper we present in the algebraic setting an intersection algorithm considered by Tworzewski [22], which is a local analytic counterpart of the Stueckrad-Vogel intersection algorithm from global algebraic geometry (cf. [20, 5)]. Some other local algebraic counterparts have been investigated by Achilles and Manaresi [1, 2]. The main purpose is to provide in this analytic context the significant method of deformation to the normal cone, which is one of the most powerful tools of intersection theory (cf. [4, 5, 2, 10]). For some important refinements and applications of this method we refer the reader to our next article [11].