The Chern-Schwartz-MacPherson class of an embeddable scheme

TL;DR

This paper generalizes a known explicit formula for the Chern-Schwartz-MacPherson class from hypersurfaces to arbitrary embeddable schemes, linking it to Segre classes of associated subschemes.

Contribution

It introduces a new explicit formula for the Chern-Schwartz-MacPherson class of embeddable schemes using Segre classes of associated subschemes, extending previous hypersurface results.

Findings

Explicit formula for Chern-Schwartz-MacPherson class of embeddable schemes

Connection between Chern-Schwartz-MacPherson class and Segre class of associated subscheme

Special case yields a formula for the Milnor class of local complete intersections

Abstract

There is an explicit formula expressing the Chern-Schwartz-MacPherson class of a hypersurface in a nonsingular variety (in characteristic $0$) in terms of the Segre class of its jacobian subscheme; this has been known for a number of years. We generalize this formula to arbitrary embeddable schemes: for every subscheme $X$ of a nonsingular variety $V$, we define an associated subscheme $Y$ of a projective bundle over $V$ and provide an explicit formula for the Chern-Schwartz-MacPherson class of $X$ in terms of the Segre class of $Y$. If $X$ is a local complete intersection, a version of the result yields a direct expression for the Milnor class of $X$.