Tensored Segre classes

TL;DR

This paper introduces a new class derived from Segre classes that simplifies intersection calculations and demonstrates its properties, including a rationality result for a related zeta function.

Contribution

It defines a tensorized Segre class incorporating line bundle data, providing simplified formulas and a rationality proof for the Segre zeta function.

Findings

Simplified formulas for intersection-theoretic calculations.

Proved rationality of the Segre zeta function for certain ideals.

Derived behavior of the class under linear joins.

Abstract

We study a class obtained from the Segre class $s(Z,Y)$ of an embedding of schemes by incorporating the datum of a line bundle on $Z$. This class satisfies basic properties analogous to the ordinary Segre class, but leads to remarkably simple formulas in standard intersection-theoretic situations such as excess or residual intersections. We prove a formula for the behavior of this class under linear joins, and use this formula to prove that a `Segre zeta function' associated with ideals generated by forms of the same degree is a rational function.