Chern classes of splayed intersections

TL;DR

This paper extends Chern class relations to possibly singular splayed intersections, providing formulas for various Chern classes and exploring conditions involving very ample divisors.

Contribution

It introduces a generalized Chern class relation for splayed intersections of singular varieties, including formulas for Segre classes and special cases involving ample divisors.

Findings

The relation holds for Chern-Schwartz-MacPherson and Chern-Fulton classes.

A formula for Segre classes of splayed subschemes is established.

The relation is analyzed under the assumption of a very ample divisor.

Abstract

We generalize the Chern class relation for the transversal intersection of two nonsingular varieties to a relation for possibly singular varieties, under a 'splayedness' assumption. The relation is shown to hold for both the Chern-Schwartz-MacPherson class and the Chern-Fulton class. The main tool is a formula for Segre classes of splayed subschemes. We also discuss the Chern class relation under the assumption that one of the varieties is a general very ample divisor.