Segre Class of Schemes with Regularly Embedded Components

TL;DR

This paper extends Fulton's Residual Intersection Theorem to compute Segre classes of schemes with regularly embedded components, providing explicit formulas based on normal bundle Chern classes and intersection properties.

Contribution

It introduces new formulas for Segre classes in schemes with regularly embedded components, generalizing existing theorems and covering cases with transverse intersections and specific blowup conditions.

Findings

Formulas for Segre classes with transversely intersecting components

Explicit expressions involving normal bundle Chern classes

Application to schemes after blowup along a component

Abstract

We generalize Fulton's Residual Intersection Theorem for the Segre class and express the Segre classes of schemes with regularly embedded components in terms of the Chern classes of the normal bundles to the components and their intersections. More specifically, we provide formulas for the following situations: when the components of the scheme intersect transversely and when the ideal sheaf of the scheme, after the blowup along a component, is the product of the ideal sheaves of the exceptional divisor and the residual scheme.