Chern classes of blow-ups

TL;DR

This paper generalizes Porteous's formula for Chern classes to blow-ups of possibly singular varieties along regularly embedded centers, simplifying proofs and exploring alternative methods involving Riemann-Roch and Chern-Schwartz-MacPherson classes.

Contribution

It extends classical formulas to singular cases and introduces a conceptually simpler proof using Riemann-Roch without denominators.

Findings

Generalized Porteous's formula to singular varieties

Provided alternative proofs using Chern-Schwartz-MacPherson classes

Simplified the proof process with explicit ideal computations

Abstract

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).