  The moduli space of stable curves of genus $g$ with $n$ marked points,
$\overline{\rm{M}}_{g,n}$, is a central object in algebraic geometry, and plays
a crucial role in $2$-dimensional conformal field theory. In this paper, we
apply the sheaf of coinvariants and conformal block divisors to study the
geometry of $\overline{\rm{M}}_{0,n}$. The main theorem characterizes the line
bundles on certain contractions of $\overline{\rm{M}}_{0,n}$ via F-curves,
using Fakhruddin's basis for the Picard group given by conformal block
divisors. This reveals a distinguished property of Knudsen's construction of
$\overline{\rm{M}}_{0,n}$. As a notable consequence of this property, using the
global generation of sheaves of coinvariants by Fakhruddin, we refine Knudsen's
construction by describing every possible contraction of
$\overline{\rm{M}}_{0,n}$ over it. An application of this refinement is given,
and its potential for generalization is discussed.
