manuscripta math. 92,273 - 286 (1997) nanuscr£pta mathematJ.ca g: Sprin~r-Yedag 199"/ Intersections of Projective Varieties and Generic Projections Hubert Flenner and Mirella Manaresi * Received August 18, 1995; in revised form November 20, 1996 Let X, Y C P~ be closed subvarieties of dimensions n and m respectively. Proving a Bezout theorem for improper intersections Stiickrad and Vogel [SVo] introduced cycles vk "=vk(X, Y) of dimension k on XNY and/~k on the ruled join variety J := J(X, Y) of X and Y which are obtained by a simple algorithm..In this paper we give an interpretation of these cycles in terms of generic projections Pk : pN ~ pn+m-k-l. For this we introduce a relative ramification locus R(Pk, X, Y) of Pk which is of dimension at most k and generalizes the usual ramification cycle in the case X = Y. We prove that this cycle is just Vk for 0 < k < dimXCIY- 1. Moreover, the cycles flk+l (for -1 < k < dimXCtY- 1) may be interpreted geometrically as the cycle of double points of Pk associated to the closure of the set of all (x : y) in the ruled join J such that (pk(x) : Pk(Y)) is in the diagonal A~,+~_k_: of j(pn+m-k-1, p'n+m-k-1). 1 Introduction Let X, Y be closed subvarieties of the projective space pN = p~ where K is an arbitrary field. Proving a Bezout theorem for improper intersections Stiickrad and Vogel [SVo] introduced cycles vk = vk(X, Y) of dimension k on X Ct Y which are obtained by a simple algorithm on the ruled join variety J:=J(X,Y):={(x:y)eP2n+~: xEX, yEY} in the following way. Let A := {(x : x) E p2N+l : x e pN} be the diagonal, so that A is given by the equations Xo-Yo ..... X~-Y~ =0, where X0,..., XN, Y0 .... , Y~, are homogeneous coordinates in p2N+l. For indeter- minates Uij (0 < i,j < N) let L be the pure transcendental extension K(Uq)o<i~i<N and let Hi c JL := J ®t¢L be the divisor given by the equation t~ := z uij(xj - ~) = o. Then one defines cycles flk and vk inductively by setting /~dim Z := [J]. If flk is already defined, decompose the intersection ~kNHdimd_k='Ok_l-I-[Jk_l (1 < k < dim J), *This paper was supported by the SFB 170 "Geometry and analysis" at the University of G6ttingen. The second named author would like to thank this institution for its hospitality