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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 forhomologically trivial algebraic cycles on $X$. Then for an embedding $f: C\toS$, we define an arithmetic diagonal cycle modified from the graph of $f$. Thiswork extends previous work of Gross and Schoen when $S$ is the product of twocurves, and is based on our recent work which relates the height pairings andthe standard conjectures.

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