  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.
