Height pairings for algebraic cycles on the product of a curve and a surface

TL;DR

This paper constructs a Beilinson--Bloch type height pairing for algebraic cycles on the product of a curve and a surface over a number field, extending previous work and relating height pairings to standard conjectures.

Contribution

It introduces an unconditional height pairing for cycles on the product of a curve and a surface, and defines an arithmetic diagonal cycle from a curve embedding, expanding prior results.

Findings

Constructed an unconditional height pairing for algebraic cycles.

Extended Gross and Schoen's work to surface products.

Linked height pairings with standard conjectures.

Abstract

For the product $X=C\times S$ of a curve and a surface over a number field, we construct unconditionally a Beilinson--Bloch type height pairing for homologically trivial algebraic cycles on $X$. Then for an embedding $f: C\to S$, we define an arithmetic diagonal cycle modified from the graph of $f$. This work extends previous work of Gross and Schoen when $S$ is the product of two curves, and is based on our recent work which relates the height pairings and the standard conjectures.