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Let $\mathcal{M}_X$ denote the moduli space of rank one logarithmic connections singular over a finite subset $S$ of a compact Riemann surface $X$ with fixed residues. We study the rational functions into $\mathcal{M}_X$. We prove that there is a natural compactification of $\mathcal{M}_X$ and the Picard group of $\mathcal{M}_X$ is isomorphic to the Picard group of $\mathrm{Pic}^d(X)$.