Let $X$ be a smooth (complex) threefold and $\gamma\in {\rm CH}_1(X)$ a homologically trivial $1$-cycle. Is there a way to construct a (singular) surface $S\subset X$ supporting $\gamma$ such that, denoting $\tau:\widetilde S\rightarrow S$ a desingularization, there is a homologically trivial $\widetilde\gamma\in Pic^0(\widetilde S)$ satisfaying $$i_*\tau_*\widetilde \gamma =\gamma \ \ {\rm in\ CH}_1(X)$$ where $i:S\hookrightarrow X$ is the inclusion?

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