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We prove that for any rationally connected threefold $X$, there exists asmooth projective surface $S$ and a family of $1$-cycles on $X$ parameterizedby $S$, inducing an Abel-Jacobi isomorphism ${\rm Alb}(S)\cong J^3(X)$. Thisstatement was previously known for some classes of smooth Fano threefolds.

Geometric representability of 1-cycles on rationally connected threefolds