An algebraic correspondence with applications to projective bundles and blowing up Chern classes

Abstract

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.

A formula for the Chern classes of symplectic blow-ups

“…Together with equation (8) this means that we have proved the formula in Theorem 9 up to an extra term f * i ! (β) on the right-hand side.…”

“…In the algebraic setting, the Chern classes of the blown-up variety are given by a 'blow-up formula' found by Porteous [13]. An alternative proof is due to Lascu and Scott [8]; cf. also [7,9].…”

Topology of Blow-ups and Enumerative Geometry

“…In algebraic geometry the formula for the Chern class of a blow-up of a nonsingular variety was first conjectured by J. A. Todd and B. Segre [29,24], confirmed by I. R. Porteous and Lascu-Scott [22,18,19]. It has been generalized to the blow ups of possibly singular varieties along regularly embedded centers by Aluffi [2].…”

A Barth–lefschetz Theorem for Submanifolds of a Product of Projective Spaces

“…Note that this construction has been already used in algebraic geometry in various circumstances, e.g. by Lascu and Scott in [17] to determine the behaviour of Chern classes undergoing a blowing up, by Deligne in [8] to simplify Fulton-Hansen connectedness theorem [10], and by the first named author in [5] to prove Lefschetz-type results for proper intersections. In the projective space P m+n+1 := Proj(k[x 0 , .…”