Line bundles on the moduli space of parabolic connections over a compact Riemann surface

Let $X$ be a compact Riemann surface of genus $g \geq 3$ and $S$ a finite subset of $X$. Let $\xi$ be fixed a holomorphic line bundle over $X$ of degree $d$. Let $\mathcal{M}_{pc}(r, d, \alpha)$ (respectively, $\mathcal{M}_{pc}(r, \alpha, \xi)$ ) denote the moduli space of parabolic connections of rank $r$, degree $d$ and full flag rational generic weight system $\alpha$, (respectively, with the fixed determinant $\xi$) singular over the parabolic points $S \subset X$. Let $\mathcal{M}'_{pc}(r, d, \alpha)$ (respectively, $\mathcal{M}'_{pc}(r, \alpha, \xi)$) be the Zariski dense open subset of $\mathcal{M}_{pc}(r, d, \alpha)$ (respectively, $\mathcal{M}_{pc}(r, \alpha, \xi)$ )parametrizing all parabolic connections such that the underlying parabolic bundle is stable. We show that there is a natural compactification of the moduli spaces $\mathcal{M}'_{pc}(r, d, \alpha)$, and $\mathcal{M}'_{pc}(r, \alpha, \xi)$ by smooth divisors. We describe the numerically effectiveness of these divisors at infinity. We determine the Picard group of the moduli spaces $\mathcal{M}_{pc}(r, d, \alpha)$, and $\mathcal{M}_{pc}(r, \alpha, \xi)$. Let $\mathcal{C}(L)$ denote the space of holomorphic connections on an ample line bundle $L$ over the moduli space $\mathcal{M}(r, d, \alpha)$ of parabolic bundles. We show that $\mathcal{C}(L)$ does not admit any non-constant algebraic function.