JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. The Johns Hopkins University Press is collaborating with JSTOR to digitize, preserve and extend access to American Journal of Mathematics. Notation and Conventions. All varieties involved are quasi-projective varieties over an algebraically closed field of arbitrary characteristic. The equivalence relation used is the rational equivalence in Chow rings.Let E be the normal bundle (of rank r=codimx Y) N(Y,X) of Y in X. Then Y' is a copy of the projective bundle P(E) whose projection is PE and we Manuscript

We shall consider exclusively algebraic non-singular quasi-projective irreducible varieties over an algebraically closed field. If V is such a variety will be the Chow ring of rational equivalence classes of cycles of Vand the group homomorphism defined by any proper morphism φ: V1 → V2. Alsodenotes the ring homomorphism defined by φ.

New techniques are developed, based on the consideration of the projective bundle associated with a direct sum o/two vector bundles, to give a simpler solution el the problem o/blowing up Chern classes which was previously solved by Porteous [12] using the Grothen. diecls Riemann-Roch theorem. Dedication.It is a particular pleasure to the authors of this work to be allowed to dedicate it, on this happy jubilee, to Beniamino SEGRE. We deal with a problem which S~ZGm~ and TODD brought to birth and to which S~GR~ has made illuminating contributions over the years, and we hope that this paper will make the solution of the problem somewhat more accessible to classical algebraic geometers.But we owe far more to S~g~ ~hau the vital scientific inspiration. It was he who brought us together and caused our paths to cross, and without him our collaboration would never have been undertaken. We both thank him from the depths of our hearts for his deeplyvalued friendship, axed for all he has done for both of us and for so many of our friends and oolleagaes in such widely separated lands.O. -Introduction.The problem of blowing up Chern classes is to compare the Chern classes of a variety X with those of the variety X' obtained by blowing up X along a subvariety :V. More precisely, if f: X'--> X is the (~ blowing down ~) morphism we have to calculate the difference between the total Chern class (or the Chern polynomial) of X' and the pull-back by f of that of X.This problem was first propounded (in terms of canonical systems) by TODD [18,19,21] and further discussed by him in the essential survey article [22]. It was also discussed by SEG~E [17] and neatly reformulated by VAT DE VE~ [23]. The first solution in general terms was given by P0~TEOUS [12]: the proof depended on the Grothendieck Riemann-Roeh Theorem, and was therefore fully effective only in characteristic zero. In positive characteristic it gave the result only modulo torsion. This latter reservation has now been implicitly removed by JOUA~OLOU [11] (*) Entrata in Redazione il 2 aprile 1973.