Invent. math. 103, 197-221 (1991) Inventiones mathematicae 9 Springer-Verlag 1991 Excess intersections and a correspondence principle Leendert J. van Gastel Rijksuniversiteit Utrecht, MathematischInstituut, Postbus 80010, NL-3508 TA Utrecht, The Netherlands Oblatum 18-X-1989 & 26-/11-1990 Introduction Recently there has been some interest in excess intersections. This subject is treated in the intersection theory of Fulton and MacPherson [6]. Stiickrad and Vogel [-18] have developed a theory for excess intersections in projective space, amounting to a generalization of Brzout's theorem. Philippon [15] has given a Brzout inequality for excessive intersections in multihomogeneous pro- jective space. From the nineteenth century stems a correspondence principle of Pieri [-16], which applied to correspondences of the form X• Y gives an approach to excess intersections. The original aim of our work, which is a part of [-9], was to find the relations between these approaches. However, the outcome was more than expected. We were not only able to give one unified theory of intersections with a family of linearly equivalent divisors that incorporates the relevant part of the general theory of Fulton and MacPherson, and all of the other theories mentioned above. But we were also led to a general correspondence principle, valid without any restriction on the dimension of the correspondence. It generalizes not only previous correspondence principles and Brzout theorems, but also well-known secant, double point and imbedding obstruction formulas and it gives new fixed point statements. In the first section we give a geometric description of the intersection algo- rithm of StiJckrad and Vogel, generalized to the setting of an arbitrary scheme Y, a morphism f: V~ Y from a pure-dimensional scheme V to Y, and a set @ = {D1 ..... Da} of linearly equivalent Cartier-divisors on Y. It produces a cycle ~ ~ V on the intersection W= (D1 c~... c~ Da) • r V, which we will call the Vogel cycle. We will briefly describe the algorithm. The D~ define a linear system and we choose d different generic elements D'~..... D~ of this system, adjoining dz indeterminates u~j to the ground field K for this purpose. (to start) Decompose [-V]=~~ ~ where ~0 is the part of the fundamental cycle [-V] supported by W, and po is the rest. (induction step) Consider the intersection of pj-i with the pull-back of the divisor D~, which we will show to be proper. Decompose it, f* O~. p j-1 =~J + pJ,