Towards a simple characterization of the Chern-Schwartz-MacPherson class

TL;DR

This paper presents a unified formula for the Chern-Schwartz-MacPherson class of certain singular varieties, potentially simplifying its computation without relying on resolution of singularities.

Contribution

It introduces a new formula for the Chern-Schwartz-MacPherson class applicable to a broad class of singular complete intersections, extending previous results and conjecturing broader validity.

Findings

Formula recovers Aluffi's 1996 hypersurface result

Proposes a conjecture for all closed subschemes of smooth varieties

Interprets the class as a Chern-Fulton class of an $\

Abstract

For a large class of possibly singular complete intersections we prove a formula for their Chern-Schwartz-MacPherson classes in terms of a single blowup along a scheme supported on the singular loci of such varieties. In the hypersurface case our formula recovers a formula of Aluffi proven in 1996. As our formula is in no way tailored to the complete intersection hypothesis, we conjecture that it holds for all closed subschemes of a smooth variety. If in fact true, such a formula would provide a simple characterization of the Chern-Schwartz-MacPherson class which does not depend on a resolution of singularities. We also show that our formula may be suitably interpreted as the Chern-Fulton class of a scheme-like object which we refer to as an `$\mathfrak{f}$-scheme'.