On CSM classes via Chern-Fulton classes of f-schemes

TL;DR

This paper introduces f-schemes, a new formal framework that extends Chern-Fulton classes to recover singularity invariants of varieties, enabling computation of Chern-Schwartz-Macpherson classes for many complete intersections.

Contribution

It proposes f-schemes as a novel concept to connect Chern-Fulton classes with singularity invariants, broadening the applicability of these classes.

Findings

Chern-Schwartz-Macpherson classes can be computed via Chern-Fulton classes of f-schemes.

F-schemes allow capturing singularity invariants not accessible through traditional schemes.

The approach applies to almost all global complete intersections.

Abstract

The Chern-Fulton class is a generalization of Chern class to the realm of arbitrary embeddable schemes. While Chern-Fulton classes are sensitive to non-reduced scheme structure, they are not sensitive to possible singularities of the underlying support, thus at first glance are not interesting from a singularity theory viewpoint. However, we introduce a class of formal objects which we think of as `fractional schemes', or f-schemes for short, and then show that when one broadens the domain of Chern-Fulton classes to f-schemes one may indeed recover singularity invariants of varieties and schemes. More precisely, we show that for almost all global complete intersections, their Chern-Schwartz-Macpherson classes may be computed via Chern-Fulton classes of f-schemes.