Properties of the Beilinson Height Pairing

TL;DR

This paper investigates the properties of the Beilinson height pairing, extending its understanding to higher dimensions and establishing its relation to intersection products through a new pairing and a projection formula.

Contribution

It introduces a new height pairing related to Beilinson's original, applicable in higher dimensions, and proves it satisfies a projection formula similar to intersection products.

Findings

Established a new height pairing linked to Beilinson's original.

Proved the pairing satisfies a projection formula.

Connected the pairing to intersection product properties.

Abstract

First constructed by Beilinson for curves defined over an algebraically closed field; R\"ossler and Szamuely generalized Beilinson's height pairing to the higher dimensional setting. In this paper we study this pairing and relate it to the intersection product by constructing a new height pairing. We also prove that it satisfies a projection formula matching the one satisfied by the intersection product.