Remarks on the generalized index of an analytic improper intersection

Abstract

Abstract. This article continues the investigation of the analytic intersection algorithm from the perspective of deformation to the normal cone, carried out in the previous papers of the author [7,8,9]. The main theorem asserts that, given an analytic set V and a linear subspace S, every collection of hyperplanes, admissible with respect to an algebraic bicone B, realizes the generalized intersection index of V and S. This result is important because the conditions for a collection of hyperplanes to be admissible with respect to B are of geometric nature: it is not necessary to analyse the embedded components of the intersections involved, but only the supports of the intersections of B with successive hyperplanes. Introduction.The generalized index at a point P of improper intersection of a purely dimensional analytic set V with a submanifold S was defined by Tworzewski [10] as the extended multiplicity at P of the total result of the analytic intersection algorithm performed for a generic admissible collection of smooth divisors. It is, of course, possible to confine oneself to the case where S is a linear subspace (flattening of S).In the papers [8,9] we reduced the computation of the generalized index at a point P of improper intersection of a purely dimensional analytic set V with a submanifold S