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We explore algebro-geometric properties of the moduli space of holomorphicLie algebroid ($\cat{L}$) connections on a compact Riemann surface $X$ of genus$g \geq 3$. We establish a smooth compactification for the moduli space of$\mathcal{L}$-connections such that underlying vector bundle is stable. Thecomplement of the moduli space into its compactification yields a divisor. Wegive a criterion for the numerical effectiveness of the divisor. We compute thePicard group of the moduli space, and analyze Lie algebroid Atiyah bundlesassociated with an ample line bundle. This enables us to conclude that regularfunctions on the space of certain Lie algebroid connections are constants.Moreover, under some condition, we show that the moduli space of$\mathcal{L}$-connections does not admit non-constant algebraic functions. Wealso explore rational connectedness of the moduli spaces.

Line bundles on the moduli space of Lie algebroid connections over a curve