The tangent direction bundle of an algebraic variety and generalized Jacobians of linear systems

Abstract

It is well-known that, on an algebraic variety V of dimension d, there is associated with a set of linear systems whose total dimension is d a Jacobian variety (of dimension d−1) at any point of which (other than base points of the linear systems) there is at least one line (formally) tangent to every variety of each system which passes through the point. This notion generalizes to a set of linear systems of total dimension d+r (0≤r<d), the generalized Jacobian being then of dimension d−r−1. The final aim of this paper is to obtain a general formula (Theorem 5.2) for the homology class of this generalized Jacobian. The proof is derived with the aid of cohomological and bundle-theoretic methods from the study of the tangent direction bundle of V, and the earlier part of the paper establishes the necessary techniques (which are not without their independent interest) for our purposes.