Chern classes of blow-ups

Details Chern classes of blow-ups Chern classes of blow-ups

2008-09-14

arxiv.org/abs/0809.2425v1

Math. Proc. Cambridge Philos. Soc. 148 (2010), no. 2, 227-242

Mathematics

Algebraic Geometry

17 pages

Scientific paper

10.1017/S0305004109990247

We extend the classical formula of Porteous for blowing-up Chern classes to the case of blow-ups of possibly singular varieties along regularly embedded centers. The proof of this generalization is perhaps conceptually simpler than the standard argument for the nonsingular case, involving Riemann-Roch without denominators. The new approach relies on the explicit computation of an ideal, and a mild generalization of the well-known formula for the normal bundle of a proper transform. We also discuss alternative, very short proofs of the standard formula in some cases: an approach relying on the theory of Chern-Schwartz-MacPherson classes (working in characteristic 0), and an argument reducing the formula to a straightforward computation of Chern classes for sheaves of differential 1-forms with logarithmic poles (when the center of the blow-up is a complete intersection).

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