A Cancellation Theorem for Segre Classes

TL;DR

This paper establishes a cancellation theorem for Segre classes, allowing the computation of the Segre class of a sub-scheme within another by relating it to a larger ambient space, with applications to generalized Riemann Kempf formulas.

Contribution

The paper introduces a new cancellation theorem for Segre classes under certain local tubular neighborhood conditions, extending the computational tools in intersection theory.

Findings

Derived a formula relating Segre classes across nested schemes

Extended Riemann Kempf formula to arbitrary integral curves

Provided conditions for formal local verification of tubular neighborhoods

Abstract

Suppose $X$ is a closed sub-scheme of $Y$ and $Y$ is a closed sub-scheme of $Z$ that formally locally has an analog of a tubular neighborhood in a sense that we define in the paper. In this setting, we prove a formula for calculating the Segre class of $X$ in $Y$ in terms of the Segre class of $X$ in $Z$ and the Chern class of the normal bundle of $Y$ in $Z$. Intuitively, this means that we can obtain the Segre class of $X$ in $Y$ by first calculating the Segre class of $X$ in $Z$, and then "cancelling out" the contribution of the embedding of $Y$ in $Z$. It is important to note that the tubular neighborhood condition may be verified formally locally. As an application, we obtain a generalization of the Riemann Kempf formula to arbitrary integral curves.